4.8: Multiple Integration (Exercises)

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4.1: Iterated Integrals and Area

In Exercises 1-6, a graph of a planar region $$R$$ is given. Give the iterated integrals, with both orders of integration $$dy\,dx$$ and $$dx\,dy$$, that give the area of $$R$$. Evaluate one of the iterated integrals to find the area.

1.

$$\text{Area} = \displaystyle \quad 9 \,\text{units}^2$$

2.

3.

$$\text{Area} = \displaystyle \quad 4 \,\text{units}^2$$

4.

5.

$$\text{Area} = \displaystyle \quad \dfrac{7}{15} \,\text{units}^2$$

6.

In Exercises 7-12, iterated integrals are given that compute the area of a region R in the $$xy$$-plane. Sketch the region R, and give the iterated integral(s) that give the area of R with the opposite order of integration.

7. $$\displaystyle \int_{-2}^2 \int_0^{4-x^2}\,dy\,dx$$

$$\displaystyle \int_{-2}^2 \int_0^{4-x^2}\,dy\,dx\quad$$ $$=\quad \displaystyle \int_{0}^4 \int_{-\sqrt{4-y}}^{\sqrt{4-y}}\,dx\,dy$$

8. $$\displaystyle \int_{0}^1 \int_{5-5x}^{5-5x^2}\,dy\,dx$$

9. $$\displaystyle \int_{-2}^2 \int_0^{2\sqrt{4-y^2}}\,dx\,dy$$

$$\displaystyle \int_{-2}^2 \int_0^{2\sqrt{4-y^2}}\,dx\,dy\quad$$ $$=\quad \displaystyle \int_{0}^4 \int_{-\frac{1}{2}\sqrt{16-x^2}}^{\frac{1}{2}\sqrt{16-x^2}}\,dy\,dx$$

10. $$\displaystyle \int_{-3}^3 \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}}\,dy\,dx$$

11. $$\displaystyle \int_{0}^1 \int_{-\sqrt{y}}^{\sqrt{y}}\,dx\,dy +\int_1^4 \int_{y-2}^{\sqrt{y}}\,dx\,dy$$

$$\displaystyle \int_{0}^1 \int_{-\sqrt{y}}^{\sqrt{y}}\,dx\,dy +\int_1^4 \int_{y-2}^{\sqrt{y}}\,dx\,dy \quad$$ $$=\quad \displaystyle \int_{-1}^2 \int_{x^2}^{x+2}\,dy\,dx$$

12. $$\displaystyle \int_{-1}^1 \int_{(x-1)/2}^{(1-x)/2}\,dy\,dx$$

In exercises 13 - 18, evaluate the iterated integrals by choosing the order of integration.

13) $$\displaystyle \int_0^{\pi} \int_0^{\pi/2} \sin(2x)\cos(3y)\,dx \space dy$$

$$0$$

14) $$\displaystyle \int_{\pi/12}^{\pi/8}\int_{\pi/4}^{\pi/3} [\cot x + \tan(2y)]\,dx \space dy$$

15) $$\displaystyle \int_1^e \int_1^e \left[\frac{1}{x}\sin(\ln x) + \frac{1}{y}\cos (\ln y)\right] \,dx \space dy$$

$$(e − 1)(1 + \sin 1 − \cos 1)$$

16) $$\displaystyle \int_1^e \int_1^e \frac{\sin(\ln x)\cos (\ln y)}{xy} \,dx \space dy$$

17) $$\displaystyle \int_1^2 \int_1^2 \left(\frac{\ln y}{x} + \frac{x}{2y + 1}\right)\,dy \space dx$$

$$\frac{3}{4}\ln \left(\frac{5}{3}\right) + 2 (\ln 2)^2 - \ln 2$$

18) $$\displaystyle \int_0^1 \int_1^2 xe^{x+4y}\,dy \space dx$$

$$\frac{1}{4}e^4 (e^4 - 1)$$

19. $$\displaystyle \int_{-3}^2 \int_2^5 (6x^2+4xy-3y^2)\,dy\,dx$$

$$\displaystyle \quad -480$$

20. $$\displaystyle \int_1^2 \int_1^x (x^2y-y+2)\,dy\,dx$$

$$\displaystyle \quad \dfrac{34}{15}$$

21. $$\displaystyle \int_0^{\pi} \int_0^{y} (\cos x \sin y)\,dx\,dy$$

$$\displaystyle \quad \dfrac{\pi}{2}$$

In exercises 22 - 25, find the average value of the function over the given rectangles.

22)$$f(x,y) = −x +2y$$, $$R = [0,1] \times [0,1]$$

$$\frac{1}{2}$$

23) $$f(x,y) = x^4 + 2y^3$$, $$R = [1,2] \times [2,3]$$

24) $$f(x,y) = \sinh x + \sinh y$$, $$R = [0,1] \times [0,2]$$

$$\frac{1}{2}(2 \space \cosh 1 + \cosh 2 - 3)$$.

25) $$f(x,y) = \arctan(xy)$$, $$R = [0,1] \times [0,1]$$

4.2: Double Integration and Volume

1) The region $$D$$ bounded by $$y = x^3, \space y = x^3 + 1, \space x = 0,$$ and $$x = 1$$ as given in the following figure.

a. Classify this region as vertically simple (Type I) or horizontally simple (Type II).

Type:
Type I but not Type II

b. Find the area of the region $$D$$.

c. Find the average value of the function $$f(x,y) = 3xy$$ on the region graphed in the previous exercise.

$$\frac{27}{20}$$

2) The region $$D$$ bounded by $$y = \sin x, \space y = 1 + \sin x, \space x = 0$$, and $$x = \frac{\pi}{2}$$ as given in the following figure.

a. Classify this region as vertically simple (Type I) or horizontally simple (Type II).

Type:
Type I but not Type II

b. Find the area of the region $$D$$.

$$\frac{\pi}{2}\, \text{units}^2$$

c. Find the average value of the function $$f(x,y) = \cos x$$ on the region $$D$$.

3) The region $$D$$ bounded by $$x = y^2 - 1$$ and $$x = \sqrt{1 - y^2}$$ as given in the following figure.

a. Classify this region as vertically simple (Type I) or horizontally simple (Type II).

Type:
Type II but not Type I

b. Find the volume of the solid under the graph of the function $$f(x,y) = xy + 1$$ and above the region $$D$$.

$$\frac{1}{6}(8 + 3\pi)\, \text{units}^3$$

4) The region $$D$$ bounded by $$y = 0, \space x = -10 + y,$$ and $$x = 10 - y$$ as given in the following figure.

a. Classify this region as vertically simple (Type I) or horizontally simple (Type II).

Type:
Type II but not Type I

b. Find the volume of the solid under the graph of the function $$f(x,y) = x + y$$ and above the region in the figure from the previous exercise.

$$\frac{1000}{3}\, \text{units}^3$$

5) The region $$D$$ bounded by $$y = 0, \space x = y - 1, \space x = \frac{\pi}{2}$$ as given in the following figure.

Classify this region as vertically simple (Type I) or horizontally simple (Type II).

Type:
Type I and Type II

6) The region $$D$$ bounded by $$y = 0$$ and $$y = x^2 - 1$$ as given in the following figure.

Classify this region as vertically simple (Type I) or horizontally simple (Type II).

Type:
Type I and Type II

7) Let $$D$$ be the region bounded by the curves of equations $$y = \cos x$$ and $$y = 4 - x^2$$ and the $$x$$-axis. Explain why $$D$$ is neither of Type I nor II.

The region $$D$$ is not of Type I: it does not lie between two vertical lines and the graphs of two continuous functions $$g_1(x)$$ and $$g_2(x)$$. The region is not of Type II: it does not lie between two horizontal lines and the graphs of two continuous functions $$h_1(y)$$ and $$h_2(y)$$.

8) Let $$D$$ be the region bounded by the curves of equations $$y = x, \space y = -x$$ and $$y = 2 - x^2$$. Explain why $$D$$ is neither of Type I nor II.

In exercises 9 - 14, evaluate the double integral $$\displaystyle \iint_D f(x,y) \,dA$$ over the region $$D$$. Graph region D.

9) $$f(x,y) = 1$$ and

$$D = \big\{(x,y)| \, 0 \leq x \leq \frac{\pi}{2}, \space \sin x \leq y \leq 1 + \sin x \big\}$$

$$\frac{\pi}{2}$$

10) $$f(x,y) = 2$$ and

$$D = \big\{(x,y)| \, 0 \leq y \leq 1, \space y - 1 \leq x \leq \arccos y \big\}$$

11) $$f(x,y) = xy$$ and

$$D = \big\{(x,y)| \, -1 \leq y \leq 1, \space y^2 - 1 \leq x \leq \sqrt{1 - y^2} \big\}$$

$$0$$

12) $$f(x,y) = \sin y$$ and $$D$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(3,0)$$

13) $$f(x,y) = -x + 1$$ and $$D$$ is the triangular region with vertices $$(0,0), \space (0,2)$$, and $$(2,2)$$

$$\frac{2}{3}$$

14) $$f(x,y) = 2x + 4y$$ and

$$D = \big\{(x,y)|\, 0 \leq x \leq 1, \space x^3 \leq y \leq x^3 + 1 \big\}$$

In exercises 15 - 20, evaluate the iterated integrals.

15) $$\displaystyle \int_0^1 \int_{2\sqrt{x}}^{2\sqrt{x}+1} (xy + 1) \,dy \space dx$$

$$\frac{41}{20}$$

16) $$\displaystyle \int_0^3 \int_{2x}^{3x} (x + y^2) \,dy \space dx$$

17) $$\displaystyle \int_1^2 \int_{-u^2-1}^{-u} (8 uv) \,dv \space du$$

$$-63$$

18) $$\displaystyle \int_e^{e^2} \int_{\ln u}^2 (v + \ln u) \,dv \space du$$

19) $$\displaystyle \int_0^{1/2} \int_{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}} 4 \,dx \space dy$$

$$\pi$$

20) $$\displaystyle \int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} (2x + 4y^3) \,dx \space dy$$

21) Let $$D$$ be the region bounded by $$y = 1 - x^2, \space y = 4 - x^2$$, and the $$x$$- and $$y$$-axes.

a. Show that $$\displaystyle \iint_D x\,dA = \int_0^1 \int_{1-x^2}^{4-x^2} x \space dy \space dx + \int_1^2 \int_0^{4-x^2} x \space dy \space dx$$ by dividing the region $$D$$ into two regions of Type I.

b. Evaluate the integral $$\displaystyle \iint_D x \,dA.$$

22) Let $$D$$ be the region bounded by $$y = 1, \space y = x, \space y = \ln x$$, and the $$x$$-axis.

a. Show that $$\displaystyle \iint_D y^2 \,dA = \int_{-1}^0 \int_{-x}^{2-x^2} y^2 dy \space dx + \int_0^1 \int_x^{2-x^2} y^2 dy \space dx$$ by dividing the region $$D$$ into two regions of Type I, where $$D = \big\{(x,y)\,|\,y \geq x, y \geq -x, \space y \leq 2-x^2\big\}$$.

b. Evaluate the integral $$\displaystyle \iint_D y^2 \,dA.$$

23) Let $$D$$ be the region bounded by $$y = x^2$$, $$y = x + 2$$, and $$y = -x$$.

a. Show that $$\displaystyle \iint_D x \, dA = \int_0^1 \int_{-y}^{\sqrt{y}} x \space dx \space dy + \int_1^4 \int_{y-2}^{\sqrt{y}} x \space dx \space dy$$ by dividing the region $$D$$ into two regions of Type II, where $$D = \big\{(x,y)\,|\,y \geq x^2, \space y \geq -x, \space y \leq x + 2\big\}$$.

b. Evaluate the integral $$\displaystyle \iint_D x \,dA.$$

b. $$\frac{7}{3}$$

24) The region $$D$$ bounded by $$x = 0, y = x^5 + 1$$, and $$y = 3 - x^2$$ is shown in the following figure. Find the area $$A(D)$$ of the region $$D$$.

25) The region $$D$$ bounded by $$y = \cos x, \space y = 4 + \cos x$$, and $$x = \pm \frac{\pi}{3}$$ is shown in the following figure. Find the area $$A(D)$$ of the region $$D$$.

$$\frac{8\pi}{3}$$

26) Find the area $$A(D)$$ of the region $$D = \big\{(x,y)| \, y \geq 1 - x^2, y \leq 4 - x^2, \space y \geq 0, \space x \geq 0 \big\}$$.

27) Let $$D$$ be the region bounded by $$y = 1, \space y = x, \space y = \ln x$$, and the $$x$$-axis. Find the area $$A(D)$$ of the region $$D$$.

$$\left(e - \frac{3}{2}\right)\, \text{units}^2$$

28) Find the average value of the function $$f(x,y) = \sin y$$ on the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(3,0)$$.

29) Find the average value of the function $$f(x,y) = -x + 1$$ on the triangular region with vertices $$(0,0), \space (0,2)$$, and $$(2,2)$$.

The average value of $$f$$ on this triangular region is $$\frac{1}{3}.$$

In exercises 30 - 33, change the order of integration and evaluate the integral.

30) $$\displaystyle \int_{-1}^{\pi/2} \int_0^{x+1} \sin x \, dy \, dx$$

31) $$\displaystyle \int_0^1 \int_{x-1}^{1-x} x \, dy \, dx$$

$$\displaystyle \int_0^1 \int_{x-1}^{1-x} x \space dy \space dx = \int_{-1}^0 \int_0^{y+1} x \space dx \space dy + \int_0^1 \int_0^{1-y} x \space dx \space dy = \frac{1}{3}$$

32) $$\displaystyle \int_{-1}^0 \int_{-\sqrt{y+1}}^{\sqrt{y+1}} y^2 dx \space dy$$

33) $$\displaystyle \int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy$$

$$\displaystyle \int_{-1/2}^{1/2} \int_{-\sqrt{y^2+1}}^{\sqrt{y^2+1}} y \space dx \space dy = \int_1^2 \int_{-\sqrt{x^2-1}}^{\sqrt{x^2-1}} y \space dy \space dx = 0$$

34)   a. $$\displaystyle \int_{0}^{4} \int_{y}^{4} e^{-x^2} \space dx \space dy$$

b. $$\displaystyle \int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{1+2y^3} \space dy \space dx$$

35) The region $$D$$ is shown in the following figure. Evaluate the double integral $$\displaystyle \iint_D (x^2 + y) \,dA$$ by using the easier order of integration.

36) The region $$D$$ is shown in the following figure. Evaluate the double integral $$\displaystyle \iint_D (x^2 - y^2) \,dA$$ by using the easier order of integration.

$$\displaystyle \iint_D (x^2 - y^2) dA = \int_{-1}^1 \int_{y^4-1}^{1-y^4} (x^2 - y^2)dx \space dy = \frac{464}{4095}$$

37) Find the volume of the solid under the surface $$z = 2x + y^2$$ and above the region bounded by $$y = x^5$$ and $$y = x$$.

38) Find the volume of the solid under the plane $$z = 3x + y$$ and above the region determined by $$y = x^7$$ and $$y = x$$.

$$\frac{4}{5}\, \text{units}^3$$

39) Find the volume of the solid under the plane $$z = 3x + y$$ and above the region bounded by $$x = \tan y, \space x = -\tan y$$, and $$x = 1$$.

40) Find the volume of the solid under the surface $$z = x^3$$ and above the plane region bounded by $$x = \sin y, \space x = -\sin y$$, and $$x = 1$$.

$$\frac{5\pi}{32}\, \text{units}^3$$

41) Let $$g$$ be a positive, increasing, and differentiable function on the interval $$[a,b]$$. Show that the volume of the solid under the surface $$z = g'(x)$$ and above the region bounded by $$y = 0, \space y = g(x), \space x = a$$, and $$x = b$$ is given by $$\frac{1}{2}(g^2 (b) - g^2 (a))$$.

42) Let $$g$$ be a positive, increasing, and differentiable function on the interval $$[a,b]$$ and let $$k$$ be a positive real number. Show that the volume of the solid under the surface $$z = g'(x)$$ and above the region bounded by $$y = g(x), \space y = g(x) + k, \space x = a$$, and $$x = b$$ is given by $$k(g(b) - g(a)).$$

43) Find the volume of the solid situated in the first octant and determined by the planes $$z = 2$$, $$z = 0, \space x + y = 1, \space x = 0$$, and $$y = 0$$.

44) Find the volume of the solid situated in the first octant and bounded by the planes $$x + 2y = 1$$, $$x = 0, \space z = 4$$, and $$z = 0$$. Graph the solid and the region of integration.

$$1\, \text{units}^3$$

45) Find the volume of the solid bounded by the planes $$x + y = 1, \space x - y = 1, \space x = 0, \space z = 0$$, and $$z = 10$$.

46) Find the volume of the solid bounded by the planes $$x + y = 1, \space x - y = 1, \space x + y = -1\space x - y = -1, \space z = 1$$, and $$z = 0$$

$$2\, \text{units}^3$$

47) Let $$S_1$$ and $$S_2$$ be the solids situated in the first octant under the planes $$x + y + z = 1$$ and $$x + y + 2z = 1$$ respectively, and let $$S$$ be the solid situated between $$S_1, \space S_2, \space x = 0$$, and $$y = 0$$.

1. Find the volume of the solid $$S_1$$.
2. Find the volume of the solid $$S_2$$.
3. Find the volume of the solid $$S$$ by subtracting the volumes of the solids $$S_1$$ and $$S_2$$.

48) Let $$S_1$$ and $$S_2$$ be the solids situated in the first octant under the planes $$2x + 2y + z = 2$$ and $$x + y + z = 1$$ respectively, and let $$S$$ be the solid situated between $$S_1, \space S_2, \space x = 0$$, and $$y = 0$$.

1. Find the volume of the solid $$S_1$$.
2. Find the volume of the solid $$S_2$$.
3. Find the volume of the solid $$S$$ by subtracting the volumes of the solids $$S_1$$ and $$S_2$$.
a. $$\frac{1}{3}\, \text{units}^3$$
b. $$\frac{1}{6}\, \text{units}^3$$
c. $$\frac{1}{6}\, \text{units}^3$$

49) Let $$S_1$$ and $$S_2$$ be the solids situated in the first octant under the plane $$x + y + z = 2$$ and under the sphere $$x^2 + y^2 + z^2 = 4$$, respectively. If the volume of the solid $$S_2$$ is $$\frac{4\pi}{3}$$ determine the volume of the solid $$S$$ situated between $$S_1$$ and $$S_2$$ by subtracting the volumes of these solids.

50) Let $$S_1$$ and $$S_2$$ be the solids situated in the first octant under the plane $$x + y + z = 2$$ and bounded by the cylinder $$x^2 + y^2 = 4$$, respectively.

1. Find the volume of the solid $$S_1$$.
2. Find the volume of the solid $$S_2$$.
3. Find the volume of the solid $$S$$ situated between $$S_1$$ and $$S_2$$ by subtracting the volumes of the solids $$S_1$$ and $$S_2$$. Graph solid S.
a. $$\frac{4}{3}\, \text{units}^3$$
b. $$2\pi\, \text{units}^3$$
c. $$\frac{6\pi - 4}{3}\, \text{units}^3$$

51) Find the volume of the solid in the first octant bounded by  $$z = \sqrt{4 - y^2}$$, $$x=2$$, $$y=2$$. Graph the solid and the region of integration.

Answer: $$2\pi$$

52) Find the volume V of the solid S bounded by the three coordinate planes, bounded above by the plane $$x+y+z=2$$, and bounded below by the plane $$z=x+y$$. Graph the solid and the region of integration.

$$\frac{1}{3}$$

53) [T] The Reuleaux triangle consists of an equilateral triangle and three regions, each of them bounded by a side of the triangle and an arc of a circle of radius s centered at the opposite vertex of the triangle. Show that the area of the Reuleaux triangle in the following figure of side length $$s$$ is $$\frac{s^2}{2}(\pi - \sqrt{3})$$.

54) [T] Show that the area of the lunes of Alhazen, the two blue lunes in the following figure, is the same as the area of the right triangle $$ABC.$$ The outer boundaries of the lunes are semicircles of diameters $$AB$$ and $$AC$$ respectively, and the inner boundaries are formed by the circumcircle of the triangle $$ABC$$.

4.3: Double Integrals in Polar Coordinates

Terms and Concepts

1. When evaluating $$\displaystyle \int\int_R f(x,y)\,dA$$ using polar coordinates, $$f(x,y)$$ is replaced with _______ and $$dA$$ is replaced with _______.

$$f(x,y)$$ is replaced with $$f(r\cos \theta, r\sin\theta)$$ and $$dA$$ is replaced with $$r\,dr\,d\theta$$.

2. Why would one be interested in evaluating a double integral with polar coordinates?

Defining Polar Regions

In exercises 3 - 6, express the region $$R$$ in polar coordinates.

3) $$R$$ is the region of the disk of radius 2 centered at the origin that lies in the first quadrant.

$$R = \big\{(r, \theta)\,|\,0 \leq r \leq 2, \space 0 \leq \theta \leq \frac{\pi}{2}\big\}$$

4) $$R$$ is the region of the disk of radius 3 centered at the origin.

5) $$R$$ is the region between the circles of radius 4 and radius 5 centered at the origin that lies in the second quadrant.

$$R = \big\{(r, \theta)\,|\,4 \leq r \leq 5, \space \frac{\pi}{2} \leq \theta \leq \pi\big\}$$

6) $$R$$ is the region bounded by the $$y$$-axis and $$x = \sqrt{1 - y^2}$$.

7) $$R$$ is the region bounded by the $$x$$-axis and $$y = \sqrt{2 - x^2}$$.

$$R = \big\{(r, \theta)\,|\,0 \leq r \leq \sqrt{2}, \space 0 \leq \theta \leq \pi\big\}$$

8) $$R = \big\{(x,y)\,|\,x^2 + y^2 \leq 4x\big\}$$

9) $$R = \big\{(x,y)\,|\,x^2 + y^2 \leq 4y\big\}$$

$$R = \big\{(r, \theta)\,|\,0 \leq r \leq 4 \space \sin \theta, \space 0 \leq \theta \leq \pi\big\}$$

In exercises 10 - 15, the graph of the polar rectangular region $$D$$ is given. Express $$D$$ in polar coordinates.

10)
11)
$$D = \big\{(r, \theta)\,|\, 3 \leq r \leq 5, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2}\big\}$$

12)

13)

$$D = \big\{(r, \theta)\,|\,3 \leq r \leq 5, \space \frac{3\pi}{4} \leq \theta \leq \frac{5\pi}{4}\big\}$$
14) In the following graph, the region $$D$$ is situated below $$y = x$$ and is bounded by $$x = 1, \space x = 5$$, and $$y = 0$$.

15) In the following graph, the region $$D$$ is bounded by $$y = x$$ and $$y = x^2$$.

$$D = \big\{(r, \theta)\,|\,0 \leq r \leq \tan \theta \space \sec \theta, \space 0 \leq \theta \leq \frac{\pi}{4}\big\}$$

Evaluating Polar Double Integrals

In exercises 16 - 25, evaluate the double integral $$\displaystyle \iint_R f(x,y) \,dA$$ over the polar rectangular region $$R$$.

16) $$f(x,y) = x^2 + y^2$$, $$R = \big\{(r, \theta)\,|\,3 \leq r \leq 5, \space 0 \leq \theta \leq 2\pi\big\}$$

17) $$f(x,y) = x + y$$, $$R = \big\{(r, \theta)\,|\,3 \leq r \leq 5, \space 0 \leq \theta \leq 2\pi\big\}$$

$$0$$

18) $$f(x,y) = x^2 + xy, \space R = \big\{(r, \theta )\,|\,1 \leq r \leq 2, \space \pi \leq \theta \leq 2\pi\big\}$$

19) $$f(x,y) = x^4 + y^4, \space R = \big\{(r, \theta)\,|\,1 \leq r \leq 2, \space \frac{3\pi}{2} \leq \theta \leq 2\pi\big\}$$

$$\frac{63\pi}{16}$$

20) $$f(x,y) = \sqrt[3]{x^2 + y^2}$$, where $$R = \big\{(r, \theta)\,|\,0 \leq r \leq 1, \space \frac{\pi}{2} \leq \theta \leq \pi\big\}$$.

21) $$f(x,y) = x^4 + 2x^2y^2 + y^4$$, where $$R = \big\{(r,\theta)\,|\,3 \leq r \leq 4, \space \frac{\pi}{3} \leq \theta \leq \frac{2\pi}{3}\big\}$$.

$$\frac{3367\pi}{18}$$

22) $$f(x,y) = \sin (\arctan \frac{y}{x})$$, where $$R = \big\{(r, \theta)\,|\,1 \leq r \leq 2, \space \frac{\pi}{6} \leq \theta \leq \frac{\pi}{3}\big\}$$

23) $$f(x,y) = \arctan \left(\frac{y}{x}\right)$$, where $$R = \big\{(r, \theta)\,|\,2 \leq r \leq 3, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{3}\big\}$$

$$\frac{35\pi^2}{576}$$

24) $$\displaystyle \iint_R e^{x^2+y^2} \left[1 + 2 \space \arctan \left(\frac{y}{x}\right)\right] \,dA, \space R = \big\{(r,\theta)\,|\,1 \leq r \leq 2, \space \frac{\pi}{6} \leq \theta \frac{\pi}{3}\big\}$$

25) $$\displaystyle \iint_R \left(e^{x^2+y^2} + x^4 + 2x^2y^2 + y^4 \right) \arctan \left(\frac{y}{x}\right) \,dA, \space R = \big\{(r, \theta )\,|\,1 \leq r \leq 2, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{3}\big\}$$

$$\frac{7}{576}\pi^2 (21 - e + e^4)$$

Converting Double Integrals to Polar Form

In exercises 26 - 29, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates.

26) $$\displaystyle \int_1^2 \int_0^x (x^2 + y^2)\,dy \space dx = \int_0^{\frac{\pi}{4}} \int_{\sec \theta}^{2 \space \sec \theta}r^3 \,dr \space d\theta$$

27) $$\displaystyle \int_2^3 \int_0^x \frac{x}{\sqrt{x^2 + y^2}}\,dy \space dx = \int_0^{\pi/4} \int_{2\sec\theta}^{3\sec \theta} \,r \space \cos \theta \space dr \space d\theta$$

$$\frac{5}{2} \ln (1 + \sqrt{2})$$

28) $$\displaystyle \int_0^1 \int_{x^2}^x \frac{1}{\sqrt{x^2 + y^2}}\,dy \space dx = \int_0^{\pi/4} \displaystyle \int_0^{\tan \theta \space \sec \theta} \space dr \space d\theta$$

29) $$\displaystyle \int_0^1 \int_{x^2}^x \frac{y}{\sqrt{x^2 + y^2}}\,dy \space dx = \int_0^{\pi/4} \displaystyle \int_0^{\tan \theta \space \sec \theta} \,r \space \sin \theta \space dr \space d\theta$$

$$\frac{1}{6}(2 - \sqrt{2})$$

In exercises 30 - 37, draw the region of integration, $$R$$, labeling all limits of integration, convert the integrals to polar coordinates and evaluate them.

30) $$\displaystyle \int_0^3 \int_0^{\sqrt{9-y^2}}\left(x^2 + y^2\right)\,dx \space dy$$

31) $$\displaystyle \int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}\left(x^2 + y^2\right)^2\,dx \space dy$$

$$\displaystyle \int_0^{\pi} \int_0^2 r^5 \,dr \space d\theta \quad = \quad \frac{32\pi}{3}$$

32) $$\displaystyle \int_0^1 \int_0^{\sqrt{1-x^2}} (x + y) \space dy \space dx$$

33) $$\displaystyle \int_0^4 \int_{-\sqrt{16-x^2}}^{\sqrt{16-x^2}} \sin (x^2 + y^2) \space dy \space dx$$

$$\displaystyle \int_{-\pi/2}^{\pi/2} \int_0^4 \,r \space \sin (r^2) \space dr \space d\theta \quad = \quad \pi \space \sin^2 8$$

34) $$\displaystyle \int_0^5 \int_{-\sqrt{25-x^2}}^{\sqrt{25-x^2}}\sqrt{x^2+y^2}\,dy\,dx$$

35) $$\displaystyle \int_{-4}^4 \int_{-\sqrt{16-y^2}}^{0}(2y-x)\,dx\,dy$$

$$\displaystyle \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} \int_0^{4} \big( 2r\sin \theta - r\cos\theta\big) \,r\,dr \space d\theta \quad = \quad \frac{128}{3}$$

36) $$\displaystyle \int_0^2 \int_{y}^{\sqrt{8-y^2}}(x+y)\,dx\,dy$$

37) $$\displaystyle \int_{-2}^{-1} \int_{0}^{\sqrt{4-x^2}}(x+5)\,dy\,dx+\int_{-1}^1\int_{\sqrt{1-x^2}}^{\sqrt{4-x^2}}(x+5)\,dy\,dx+\int_1^2\int_0^{\sqrt{4-x^2}}(x+5)\,dy\,dx$$

$$\displaystyle \int_{0}^{\pi} \int_1^{2} \big( r\cos \theta + 5\big) \,r\,dr \space d\theta \quad = \quad \frac{15\pi}{2}$$

38) Evaluate the integral $$\displaystyle \iint_D r \,dA$$ where $$D$$ is the region bounded by the polar axis and the upper half of the cardioid $$r = 1 + \cos \theta$$.

39) Find the area of the region $$D$$ bounded by the polar axis and the upper half of the cardioid $$r = 1 + \cos \theta$$.

$$\frac{3\pi}{4}$$

40) Evaluate the integral $$\displaystyle \iint_D r \,dA,$$ where $$D$$ is the region bounded by the part of the four-leaved rose $$r = \sin 2\theta$$ situated in the first quadrant (see the following figure).

41) Find the total area of the region enclosed by the four-leaved rose $$r = \sin 2\theta$$ (see the figure in the previous exercise).

$$\frac{\pi}{2}$$

42) Find the area of the region $$D$$ which is the region bounded by $$y = \sqrt{4 - x^2}$$, $$x = \sqrt{3}$$, $$x = 2$$, and $$y = 0$$.

43) Find the area of the region $$D$$, which is the region inside the disk $$x^2 + y^2 \leq 4$$ and to the right of the line $$x = 1$$.

$$\frac{1}{3}(4\pi - 3\sqrt{3})$$

44) Determine the average value of the function $$f(x,y) = x^2 + y^2$$ over the region $$D$$ bounded by the polar curve $$r = \cos 2\theta$$, where $$-\frac{\pi}{4} \leq \theta \leq \frac{\pi}{4}$$ (see the following graph).

45) Determine the average value of the function $$f(x,y) = \sqrt{x^2 + y^2}$$ over the region $$D$$ bounded by the polar curve $$r = 3\sin 2\theta$$, where $$0 \leq \theta \leq \frac{\pi}{2}$$ (see the following graph).

$$\frac{16}{3\pi}$$

46) Find the volume of the solid situated in the first octant and bounded by the paraboloid $$z = 1 - 4x^2 - 4y^2$$ and the planes $$x = 0, \space y = 0$$, and $$z = 0$$.

47) Find the volume of the solid bounded by the paraboloid $$z = 2 - 9x^2 - 9y^2$$ and the plane $$z = 1$$.

$$\frac{\pi}{18}$$

48)

1. Find the volume of the solid $$S_1$$ bounded by the cylinder $$x^2 + y^2 = 1$$ and the planes $$z = 0$$ and $$z = 1$$.
2. Find the volume of the solid $$S_2$$ outside the double cone $$z^2 = x^2 + y^2$$ inside the cylinder $$x^2 + y^2 = 1$$, and above the plane $$z = 0$$.
3. Find the volume of the solid inside the cone $$z^2 = x^2 + y^2$$ and below the plane $$z = 1$$ by subtracting the volumes of the solids $$S_1$$ and $$S_2$$.

49)

1. Find the volume of the solid $$S_1$$ inside the unit sphere $$x^2 + y^2 + z^2 = 1$$ and above the plane $$z = 0$$.
2. Find the volume of the solid $$S_2$$ inside the double cone $$(z - 1)^2 = x^2 + y^2$$ and above the plane $$z = 0$$.
3. Find the volume of the solid outside the double cone $$(z - 1)^2 = x^2 + y^2$$ and inside the sphere $$x^2 + y^2 + z^2 = 1$$.
a. $$\frac{2\pi}{3}$$; b. $$\frac{\pi}{3}$$; c. $$\frac{\pi}{3}$$

In Exercises 50-51, special double integrals are presented that are especially well suited for evaluation in polar coordinates.

50) The surface of a right circular cone with height $$h$$ and base radius $$a$$ can be described by the equation $$f(x,y)=h-h\sqrt{\frac{x^2}{a^2}+\frac{y^2}{a^2}}$$, where the tip of the cone lies at $$(0,0,h)$$ and the circular base lies in the $$xy$$-plane, centered at the origin.
Confirm that the volume of a right circular cone with height $$h$$ and base radius $$a$$ is $$V=\frac{1}{3}\pi a^2h$$ by evaluating $$\displaystyle \int\int_R f(x,y)\,dA$$ in polar coordinates.

51) Consider $$\displaystyle \int\int_R e^{-(x^2+y^2)}\,dA.$$
(a) Why is this integral difficult to evaluate in rectangular coordinates, regardless of the region $$R$$?
(b) Let $$R$$ be the region bounded by the circle of radius $$a$$ centered at the origin. Evaluate the double integral using polar coordinates.
(c) Take the limit of your answer from (b), as $$a\to \infty$$. What does this imply about the volume under the surface of $$e^{-(x^2+y^2)}$$ over the entire $$xy$$-plane?

For the following two exercises, consider a spherical ring, which is a sphere with a cylindrical hole cut so that the axis of the cylinder passes through the center of the sphere (see the following figure).

52) If the sphere has radius 4 and the cylinder has radius 2 find the volume of the spherical ring.

53) A cylindrical hole of diameter 6 cm is bored through a sphere of radius 5 cm such that the axis of the cylinder passes through the center of the sphere. Find the volume of the resulting spherical ring.

$$\frac{256\pi}{3} \space \text{cm}^3$$

54) Find the volume of the solid that lies under the double cone $$z^2 = 4x^2 + 4y^2$$, inside the cylinder $$x^2 + y^2 = x$$, and above the plane $$z = 0$$.

55) Find the volume of the solid that lies under the paraboloid $$z = x^2 + y^2$$, inside the cylinder $$x^2 + y^2 = 1$$ and above the plane $$z = 0$$.

$$\frac{\pi}{2}$$

56) Find the volume of the solid that lies under the plane $$x + y + z = 10$$ and above the disk $$x^2 + y^2 = 4x$$.

57) Find the volume of the solid that lies under the plane $$2x + y + 2z = 8$$ and above the unit disk $$x^2 + y^2 = 1$$.

$$4\pi$$

58) A radial function $$f$$ is a function whose value at each point depends only on the distance between that point and the origin of the system of coordinates; that is, $$f (x,y) = g(r)$$, where $$r = \sqrt{x^2 + y^2}$$. Show that if $$f$$ is a continuous radial function, then

$\iint_D f(x,y)dA = (\theta_2 - \theta_1) [G(R_2) - G(R_1)], \space where \space G'(r) = rg(r)$ and $$(x,y) \in D = \big\{(r, \theta)\,|\,R_1 \leq r \leq R_2, \space 0 \leq \theta \leq 2\pi\big\}$$, with $$0 \leq R_1 < R_2$$ and $$0 \leq \theta_1 < \theta_2 \leq 2\pi$$.

59) Use the information from the preceding exercise to calculate the integral $$\displaystyle \iint_D (x^2 + y^2)^3 dA,$$ where $$D$$ is the unit disk.

$$\frac{\pi}{4}$$

60) Let $$f(x,y) = \dfrac{F'(r)}{r}$$ be a continuous radial function defined on the annular region $$D = \big\{(r,\theta)\,|\,R_1 \leq r \leq R_2, \space 0 \leq \theta \leq 2\pi\big\}$$, where $$r = \sqrt{x^2 + y^2}$$, $$0 < R_1 < R_2$$, and $$F$$ is a differentiable function.

Show that $$\displaystyle \iint_D f(x,y)\,dA = 2\pi [F(R_2) - F(R_1)].$$

61) Apply the preceding exercise to calculate the integral $$\displaystyle \iint_D \frac{e^{\sqrt{x^2+y^2}}}{\sqrt{x^2 + y^2}} \,dx \space dy$$ where $$D$$ is the annular region between the circles of radii 1 and 2 situated in the third quadrant.

$$\frac{1}{2} \pi e(e - 1)$$

62) Let $$f$$ be a continuous function that can be expressed in polar coordinates as a function of $$\theta$$ only; that is, $$f(x,y) = h(\theta)$$, where $$(x,y) \in D = \big\{(r, \theta)\,|\,R_1 \leq r \leq R_2, \space \theta_1 \leq \theta \leq \theta_2\big\}$$, with $$0 \leq R_1 < R_2$$ and $$0 \leq \theta_1 < \theta_2 \leq 2\pi$$.

Show that $$\displaystyle \iint_D f(x,y) \,dA = \frac{1}{2} (R_2^2 - R_1^2) [H(\theta_2) - H(\theta_1)]$$, where $$H$$ is an antiderivative of $$h$$.

63) Apply the preceding exercise to calculate the integral $$\displaystyle \iint_D \frac{y^2}{x^2}\,dA,$$ where $$D = \big\{(r, \theta)\,|\, 1 \leq r \leq 2, \space \frac{\pi}{6} \leq \theta \leq \frac{\pi}{3}\big\}.$$

$$\sqrt{3} - \frac{\pi}{4}$$

64) Let $$f$$ be a continuous function that can be expressed in polar coordinates as a function of $$\theta$$ only; that is $$f(x,y) = g(r)h(\theta)$$, where $$(x,y) \in \big\{(r, \theta )\,|\,R_1 \leq r \leq R_2, \space \theta_1 \leq \theta \leq \theta_2\big\}$$ with $$0 \leq R_1 < R_2$$ and $$0 \leq \theta_1 < \theta_2 \leq 2\pi$$. Show that $\iint_D f(x,y)\,dA = [G(R_2) - G(R_1)] \space [H(\theta_2) - H(\theta_1)],$ where $$G$$ and $$H$$ are antiderivatives of $$g$$ and $$h$$, respectively.

65) Evaluate $$\displaystyle \iint_D \arctan \left(\frac{y}{x}\right) \sqrt{x^2 + y^2}\,dA,$$ where $$D = \big\{(r,\theta)\,|\, 2 \leq r \leq 3, \space \frac{\pi}{4} \leq \theta \leq \frac{\pi}{3}\big\}$$.

$$\frac{133}{864}\pi^2$$

66) A spherical cap is the region of a sphere that lies above or below a given plane.

a. Show that the volume of the spherical cap in the figure below is $$\frac{1}{6} \pi h (3a^2 + h^2)$$.

b. A spherical segment is the solid defined by intersecting a sphere with two parallel planes. If the distance between the planes is $$h$$ show that the volume of the spherical segment in the figure below is $$\frac{1}{6}\pi h (3a^2 + 3b^2 + h^2)$$.

67) In statistics, the joint density for two independent, normally distributed events with a mean $$\mu = 0$$ and a standard distribution $$\sigma$$ is defined by $$p(x,y) = \frac{1}{2\pi\sigma^2} e^{-\frac{x^2+y^2}{2\sigma^2}}$$. Consider $$(X,Y)$$, the Cartesian coordinates of a ball in the resting position after it was released from a position on the z-axis toward the $$xy$$-plane. Assume that the coordinates of the ball are independently normally distributed with a mean $$\mu = 0$$ and a standard deviation of $$\sigma$$ (in feet). The probability that the ball will stop no more than $$a$$ feet from the origin is given by $P[X^2 + Y^2 \leq a^2] = \iint_D p(x,y) dy \space dx,$ where $$D$$ is the disk of radius $$a$$ centered at the origin. Show that $P[X^2 + Y^2 \leq a^2] = 1 - e^{-a^2/2\sigma^2}.$

68) The double improper integral $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \, dx$ may be defined as the limit value of the double integrals $$\displaystyle \iint_D e^{-x^2+y^2/2}\,dA$$ over disks $$D_a$$ of radii $$a$$ centered at the origin, as $$a$$ increases without bound; that is,

$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}dy \space dx = \lim_{a\rightarrow\infty} \iint_{D_a} e^{-x^2+y^2/2}\,dA.$

Use polar coordinates to show that $$\displaystyle \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \, dx = 2\pi.$$

69) Show that $$\displaystyle \int_{-\infty}^{\infty} e^{-x^2/2}\,dx = \sqrt{2\pi}$$ by using the relation

$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2+y^2/2}\,dy \,dx = \left(\int_{-\infty}^{\infty} e^{-x^2/2}dx \right) \left( \int_{-\infty}^{\infty} e^{-y^2/2}dy \right).$

4.4: Triple Integrals

Terms and Concepts

1. The strategy for establishing bounds for triple integrals is "from ________ to ________, then from ________ to ________ and then from ________ to ________."

We integrate from surface to surface, then from curve to curve and then from point to point.

2. Give an informal interpretation of what $$\displaystyle \iiint_Q \,dV$$ means.

$$\displaystyle \iiint_Q \,dV$$ = Volume of the solid region $$Q$$

3. Give two uses of triple integration.

To compute total mass or average density of a solid object, given a density function or to compute the average temperature in a solid region or object.

4. If an object has a constant density $$\delta$$ and a volume $$V$$, what is its mass?

It's mass is $$\delta V$$.

Volume of Solid Regions

In Exercises 5-8, two surfaces $$f_1(x,y)$$ and $$f_2(x,y)$$ and a region $$R$$ in the $$xy$$-plane are given. Set up and evaluate the triple integral that represents the volume between these surfaces over $$R$$.

5. $$f_1(x,y) = 8-x^2-y^2,\,f_2(x,y) =2x+y;$$
$$R$$ is the square with corners $$(-1,-1)$$ and $$(1,1)$$.

V = $$\displaystyle \int_{-1}^{1}\int_{-1}^{1}\int_{2x+y}^{8-x^2-y^2} \,dz\,dy\,dx\quad$$ $$=\quad\dfrac{88}{3}\,\text{units}^3$$

6. $$f_1(x,y) = x^2+y^2,\,f_2(x,y) =-x^2-y^2;$$
$$R$$ is the square with corners $$(0,0)$$ and $$(2,3)$$.

7. $$f_1(x,y) = \sin x \cos y,\,f_2(x,y) =\cos x \sin y +2;$$
$$R$$ is the triangle with corners $$(0,0), \,(\pi , 0)$$ and $$(\pi,\pi)$$.

V = $$\displaystyle \int_{0}^{\pi}\int_{0}^{x}\int_{\sin x\cos y}^{\cos x\sin y + 2} \,dz\,dy\,dx\quad$$ $$=\quad\left(\pi^2 - \pi\right)\,\text{units}^3\quad$$ $$\approx 6.72801\,\text{units}^3$$

8. $$f_1(x,y) = 2x^2+2y^2+3,\,f_2(x,y) =6-x^2-y^2;$$
$$R$$ is the circle $$x^2+y^2=1$$.

In Exercises 9-16, a solid $$D$$ is described by its bounding surfaces. Graph the solid. Set up the triple integral that gives the volume of $$D$$ in the indicated order(s) of integration. Evaluate the triple integral to find this volume if instructed.

9. $$D$$ is bounded by the coordinate planes and $$z=2-\frac{2}{3}x-2y$$.
Evaluate the triple integral with order $$dz\,dy\,dx$$.

V = $$\displaystyle \int_{0}^{3}\int_{0}^{1-\frac{x}{3}}\int_{0}^{2 - \frac{2}{3}x-2y} \,dz\,dy\,dx\quad$$ $$=\quad 1\,\text{unit}^3$$

10. $$D$$ is bounded by the planes $$y=0,y=2,x=1,z=0$$ and $$z=(2-x)/2$$.
Evaluate the triple integral with order $$dx\,dy\,dz$$.

11. $$D$$ is bounded by the planes $$x=0,x=2,z=-y$$ and by $$z=y^2/2$$.
Evaluate the triple integral with orders $$dy\,dz\,dx$$ and $$dz\,dy\,dx$$ to verify that you obtain the same volume either way.

V = $$\displaystyle \int_{0}^{2}\int_{0}^{2}\int_{-\sqrt{2z}}^{-z} \,dy\,dz\,dx\quad$$ $$=\quad \dfrac{4}{3}\,\text{unit}^3$$
V = $$\displaystyle \int_{0}^{2}\int_{-2}^{0}\int_{\frac{y^2}{2}}^{-y} \,dz\,dy\,dx\quad$$ $$=\quad \dfrac{4}{3}\,\text{unit}^3$$

12. $$D$$ is bounded by the planes $$z=0,y=9, x=0$$ and by $$z=\sqrt{y^2-9x^2}$$.
Do not evaluate any triple integral. Just set this one up.

13. $$D$$ is bounded by the planes $$x=2,y=1,z=0$$ and $$z=2x+4y-4$$.
Evaluate the triple integral with orders $$dz\,dy\,dx$$ and $$dx\,dy\,dz$$ to verify that you obtain the same volume either way.

V = $$\displaystyle \int_{0}^{2}\int_{1-\frac{x}{2}}^{1}\int_{0}^{2x+4y-4} \,dz\,dy\,dx\quad$$ $$=\quad\dfrac{4}{3}\,\text{units}^3$$
V = $$\displaystyle \int_{0}^{4}\int_{\frac{z}{4}}^{1}\int_{(z-4y+4)/2}^{2} \,dx\,dy\,dz\quad$$ $$=\quad\dfrac{4}{3}\,\text{units}^3$$

14. $$D$$ is bounded by the plane $$z=2y$$ and by $$y=4-x^2$$.
Evaluate the triple integral with order $$dz\,dy\,dx$$.

15. $$D$$ is bounded by the coordinate planes and $$y=1-x^2$$ and $$y=1-z^2$$.
Do not evaluate any triple integral. Which order would be easier to evaluate: $$dz\,dy\,dx$$ or $$dy\,dz\,dx$$? Explain why.

V = $$\displaystyle \int_{0}^{1}\int_{0}^{1-x^2}\int_{0}^{\sqrt{1-y}} \,dz\,dy\,dx\quad$$
V = $$\displaystyle \int_{0}^{1}\int_{0}^{x}\int_{0}^{1-x^2} \,dy\,dz\,dx + \displaystyle \int_{0}^{1}\int_{x}^{1}\int_{0}^{1-z^2} \,dy\,dz\,dx$$
The first one is easier since it only requires evaluation of a single integral, although both can be evaluated fairly easily.

16. $$D$$ is bounded by the coordinate planes and by $$z=1-y/3$$ and $$z=1-x$$.
Evaluate the triple integral with order $$dx\,dy\,dz$$.

Evaluating General Triple Integrals

In exercises 17 - 20, evaluate the triple integrals over the rectangular solid box $$B$$.

17. $$\displaystyle \iiint_B (2x + 3y^2 + 4z^3) \space dV,$$ where $$B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\big\}$$

$$192$$

18. $$\displaystyle \iiint_B (xy + yz + xz) \space dV,$$ where $$B = \big\{(x,y,z) \,|\, 1 \leq x \leq 2, \space 0 \leq y \leq 2, \space 1 \leq z \leq 3\big\}$$

19. $$\displaystyle \iiint_B (x \space cos \space y + z) \space dV,$$ where $$B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq \pi, \space -1 \leq z \leq 1\big\}$$

$$0$$

20. $$\displaystyle \iiint_B (z \space sin \space x + y^2) \space dV,$$ where $$B = \big\{(x,y,z) \,|\, 0 \leq x \leq \pi, \space 0 \leq y \leq 1, \space -1 \leq z \leq 2\big\}$$

In Exercises 21 - 24, evaluate the triple integral.

21. $$\displaystyle \int_{-\pi/2}^{\pi/2}\int_{0}^{\pi}\int_{0}^{\pi} (\cos x \sin y \sin z )\,dz\,dy\,dx$$

$$8$$

22. $$\displaystyle \int_{0}^{1}\int_{0}^{x}\int_{0}^{x+y} (x+y+z )\,dz\,dy\,dx$$

23. $$\displaystyle \int_{0}^{\pi}\int_{0}^{1}\int_{0}^{z} (\sin (yz))\,dx\,dy\,dz$$

$$\pi$$

24. $$\displaystyle \int_{\pi}^{\pi^2}\int_{x}^{x^3}\int_{-y^2}^{y^2} (\cos x \sin y \sin z )\,dz\,dy\,dx$$

Average Value of a Function

25. Find the average value of the function $$f(x,y,z) = x + y + z$$ over the parallelepiped determined by $$x = 0, \space x = 1, \space y = 0, \space y = 3, \space z = 0$$, and $$z = 5$$.

$$\frac{9}{2}$$

26. Find the average value of the function $$f(x,y,z) = xyz$$ over the solid $$E = [0,1] \times [0,1] \times [0,1]$$ situated in the first octant.

Approximating Triple Integrals

27. The midpoint rule for the triple integral $$\displaystyle \iiint_B f(x,y,z) \,dV$$ over the rectangular solid box $$B$$ is a generalization of the midpoint rule for double integrals. The region $$B$$ is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum $\sum_{i=1}^l \sum_{j=1}^m \sum_{k=1}^n f(\bar{x_i}, \bar{y_j}, \bar{z_k}) \Delta V,\nonumber$ where $$(\bar{x_i}, \bar{y_j}, \bar{z_k})$$ is the center of the box $$B_{ijk}$$ and $$\Delta V$$ is the volume of each subbox. Apply the midpoint rule to approximate $\iiint_B x^2 \,dV\nonumber$ over the solid $$B = \big\{(x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 1, \space 0 \leq z \leq 1 \big\}$$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

$$\displaystyle \iiint_B f(x,y,z) \,dV\quad$$ $$\approx\quad\frac{5}{16} \approx 0.313$$

28. [T] a. Apply the midpoint rule to approximate $\iiint_B e^{-x^2} \,dV\nonumber$ over the solid $$B = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq 1, \space 0 \leq z \leq 1 \}$$ by using a partition of eight cubes of equal size. Round your answer to three decimal places.

b. Use a CAS to improve the above integral approximation in the case of a partition of $$n^3$$ cubes of equal size, where $$n = 3,4, ..., 10$$.

Applications

29. Suppose that the temperature in degrees Celsius at a point $$(x,y,z)$$ of a solid $$E$$ bounded by the coordinate planes and the plane $$x + y + z = 5$$ is given by: $T (x,y,z) = xz + 5z + 10\nonumber$ Find the average temperature over the solid.

$$17.5^{\circ}$$ C

30. Suppose that the temperature in degrees Fahrenheit at a point $$(x,y,z)$$ of a solid $$E$$ bounded by the coordinate planes and the plane $$x + y + z = 5$$ is given by: $T(x,y,z) = x + y + xy\nonumber$ Find the average temperature over the solid.

31. If the charge density at an arbitrary point $$(x,y,z)$$ of a solid $$E$$ is given by the function $$\rho (x,y,z)$$, then the total charge inside the solid is defined as the triple integral $$\displaystyle \iiint_E \rho (x,y,z) \,dV.$$ Assume that the charge density of the solid $$E$$ enclosed by the paraboloids $$x = 5 - y^2 - z^2$$ and $$x = y^2 + z^2 - 5$$ is equal to the distance from an arbitrary point of $$E$$ to the origin. Set up the integral that gives the total charge inside the solid $$E$$.

Total Charge inside the Solid $$E \quad=\quad$$ $$\displaystyle \int_{-\sqrt{5}}^{\sqrt{5}}\int_{-\sqrt{5-y^2}}^{\sqrt{5-y^2}}\int_{y^2+z^2-5}^{5 - y^2 - z^2} \sqrt{x^2+y^2+z^2}\,dx\,dz\,dy$$

32. Show that the volume of a regular right hexagonal pyramid of edge length $$a$$ is $$\dfrac{a^3 \sqrt{3}}{2}$$ by using triple integrals.

4.5: Triple Integrals in Cylindrical and Spherical Coordinates

In exercises 1 - 8, evaluate the triple integrals $$\displaystyle \iiint_E f(x,y,z) \, dV$$ over the solid $$E$$.

1. $$f(x,y,z) = z, \quad E = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 9, \quad x \leq 0, \quad y \leq 0, \quad 0 \leq z \leq 1\big\}$$

$$\frac{9\pi}{8}$$

2. $$f(x,y,z) = xz^2, \space E = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 16, \space x \geq 0, \space y \leq 0, \space -1 \leq z \leq 1\big\}$$

3. $$f(x,y,z) = xy, \space E = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 1, \space x \geq 0, \space x \geq y, \space -1 \leq z \leq 1\big\}$$

$$-\frac{1}{8}$$

4. $$f(x,y,z) = x^2 + y^2, \space E = \big\{(x,y,z)\, | \,x^2 + y^2 \leq 4, \space x \geq 0, \space x \leq y, \space 0 \leq z \leq 3\big\}$$

5. $$f(x,y,z) = e^{\sqrt{x^2+y^2}}, \space E = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 \leq 4, \space y \leq 0, \space x \leq y\sqrt{3}, \space 2 \leq z \leq 3 \big\}$$

$$\frac{\pi e^2}{6}$$

6. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space E = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 \leq 9, \space y \leq 0, \space 0 \leq z \leq 1\big\}$$

7. a. Let $$B$$ be a cylindrical shell with inner radius $$a$$ outer radius $$b$$, and height $$c$$ where $$0 < a < b$$ and $$c>0$$. Assume that a function $$F$$ defined on $$B$$ can be expressed in cylindrical coordinates as $$F(x,y,z) = f(r) + h(z)$$, where $$f$$ and $$h$$ are differentiable functions. If $$\displaystyle \int_a^b \bar{f} (r) \,dr = 0$$ and $$\bar{h}(0) = 0$$, where $$\bar{f}$$ and $$\bar{h}$$ are antiderivatives of $$f$$ and $$h$$, respectively, show that $$\displaystyle \iiint_B F(x,y,z) \,dV = 2\pi c (b\bar{f} (b) - a \bar{f}(a)) + \pi(b^2 - a^2) \bar{h} (c).$$

b. Use the previous result to show that $$\displaystyle \iiint_B \left(z + \sin \sqrt{x^2 + y^2}\right) \,dx \space dy \space dz = 6 \pi^2 ( \pi - 2),$$ where $$B$$ is a cylindrical shell with inner radius $$\pi$$ outer radius $$2\pi$$, and height $$2$$.

8. a. Let $$B$$ be a cylindrical shell with inner radius $$a$$ outer radius $$b$$ and height $$c$$ where $$0 < a < b$$ and $$c > 0$$. Assume that a function $$F$$ defined on $$B$$ can be expressed in cylindrical coordinates as $$F(x,y,z) = f(r) g(\theta) f(z)$$, where $$f, \space g,$$ and $$h$$ are differentiable functions. If $$\displaystyle\int_a^b \tilde{f} (r) \, dr = 0,$$ where $$\tilde{f}$$ is an antiderivative of $$f$$, show that $$\displaystyle\iiint_B F (x,y,z)\,dV = [b\tilde{f}(b) - a\tilde{f}(a)] [\tilde{g}(2\pi) - \tilde{g}(0)] [\tilde{h}(c) - \tilde{h}(0)],$$ where $$\tilde{g}$$ and $$\tilde{h}$$ are antiderivatives of $$g$$ and $$h$$, respectively.

b. Use the previous result to show that $$\displaystyle\iiint_B z \sin \sqrt{x^2 + y^2} \,dx \space dy \space dz = - 12 \pi^2,$$ where $$B$$ is a cylindrical shell with inner radius $$\pi$$ outer radius $$2\pi$$, and height $$2$$.

In exercises 9 - 12, the boundaries of the solid $$E$$ are given in cylindrical coordinates.

a. Graph solid E

b. Express the region $$E$$ in cylindrical coordinates.

c. Convert the integral $$\displaystyle \iiint_E f(x,y,z) \,dV$$ to cylindrical coordinates.

9. $$E$$ is bounded by the right circular cylinder $$r = 4 \sin \theta$$, the $$r\theta$$-plane, and the sphere $$r^2 + z^2 = 16$$.

a. $$E = \big\{(r,\theta,z)\, | \,0 \leq \theta \leq \pi, \space 0 \leq r \leq 4 \sin \theta, \space 0 \leq z \leq \sqrt{16 - r^2}\big\}$$

b. $$\displaystyle\int_0^{\pi} \int_0^{4 \sin \theta} \int_0^{\sqrt{16-r^2}} f(r,\theta, z) r \, dz \space dr \space d\theta$$

10. $$E$$ is bounded by the right circular cylinder $$r = \cos \theta$$, the $$r\theta$$-plane, and the sphere $$r^2 + z^2 = 9$$.

11. $$E$$ is located in the first octant and is bounded by the circular paraboloid $$z = 9 - 3r^2$$, the cylinder $$r = \sqrt{3}$$, and the plane $$r(\cos \theta + \sin \theta) = 20 - z$$.

a. $$E = \big\{(r,\theta,z) \, | \, 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq r \leq \sqrt{3}, \space 9 - r^2 \leq z \leq 10 - r(\cos \theta + \sin \theta)\big\}$$

b. $$\displaystyle\int_0^{\pi/2} \int_0^{\sqrt{3}} \int_{9-r^2}^{10-r(\cos \theta + \sin \theta)} f(r,\theta,z) r \space dz \space dr \space d\theta$$

12. $$E$$ is located in the first octant outside the circular paraboloid $$z = 10 - 2r^2$$ and inside the cylinder $$r = \sqrt{5}$$ and is bounded also by the planes $$z = 20$$ and $$\theta = \frac{\pi}{4}$$.

In exercises 13 - 16, the function $$f$$ and solid $$E$$ are given.

a. Graph solid E

b. Express the solid $$E$$ and the function $$f$$ in cylindrical coordinates.

c. Convert the integral $$\displaystyle \iiint_E f(x,y,z) \,dV$$ into cylindrical coordinates and evaluate it.

13. $$f(x,y,z) = x^2 + y^2$$, $$E = \big\{(x,y,z)\, | \,0 \leq x^2 + y^2 \leq 9, \space x \geq 0, \space y \geq 0, \space 0 \leq z \leq x + 3\big\}$$

c. $$\frac{3^5}{5}+\frac{3^5 \pi}{8}$$

14. $$f(x,y,z) = x^2 + y^2, \space E = \big\{(x,y,z) |0 \leq x^2 + y^2 \leq 4, \space y \geq 0, \space 0 \leq z \leq 3 - x \big\}$$

15. $$f(x,y,z) = x, \space E = \big\{(x,y,z)\, | \,1 \leq y^2 + z^2 \leq 9, \space 0 \leq x \leq 1 - y^2 - z^2\big\}$$

a. $$y = r \space \cos \theta, \space z = r \space \sin \theta, \space x = z,\space E = \big\{(r,\theta,z)\, | \,1 \leq r \leq 3, \space 0 \leq \theta \leq 2\pi, \space 0 \leq z \leq 1 - r^2\big\}, \space f(r,\theta,z) = z$$;

b. $$\displaystyle \int_1^3 \int_0^{2\pi} \int_0^{1-r^2} z r \space dz \space d\theta \space dr = \frac{356 \pi}{3}$$

16. $$f(x,y,z) = y, \space E = \big\{(x,y,z)\, | \,1 \leq x^2 + z^2 \leq 9, \space 0 \leq y \leq 1 - x^2 - z^2 \big\}$$

In exercises 17 - 24, find the volume of the solid $$E$$ whose boundaries are given in rectangular coordinates. Graph solid E.

17. $$E$$ is above the $$xy$$-plane, inside the cylinder $$x^2 + y^2 = 1$$, and below the plane $$z = 1$$.

$$\pi$$

18. $$E$$ is below the plane $$z = 1$$ and inside the paraboloid $$z = x^2 + y^2$$.

19. $$E$$ is bounded by the circular cone $$z = \sqrt{x^2 + y^2}$$ and $$z = 1$$.

$$\frac{\pi}{3}$$

20. $$E$$ is located above the $$xy$$-plane, below $$z = 1$$, outside the one-sheeted hyperboloid $$x^2 + y^2 - z^2 = 1$$, and inside the cylinder $$x^2 + y^2 = 2$$.

21. $$E$$ is located below  $$z = 1 - x^2 - y^2$$ and above $$z = x^2 + y^2$$.

$$\frac{\pi}{4}$$

22. $$E$$ is located inside the sphere $$x^2 + y^2 + z^2 = 1$$, above the $$xy$$-plane, and inside the circular cone $$z = \sqrt{x^2 + y^2}$$.

23. $$E$$ is located outside the circular cone $$x^2 + y^2 = (z - 1)^2$$ and inside the right circular cylinder $$x^2 + y^2 = 1$$.

$$\frac{4\pi}{3}$$

24. $$E$$ is located outside the circular cone $$z = 1 - \sqrt{x^2 + y^2}$$, above the $$xy$$-plane, below the circular paraboloid, and between the planes $$z = 0$$ and $$z = 2$$.

25. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $$\displaystyle \int_{-\pi/2}^{\pi/2} \int_0^1 \int_{r^2}^r r \, dz \, dr \, d\theta.$$  Find the volume $$V$$ of the solid. Round your answer to four decimal places.

$$V = \frac{pi}{12} \approx 0.2618$$

26. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates $$\displaystyle \int_0^{\pi/2} \int_0^1 \int_{r^4}^r r \, dz \, dr \, d\theta.$$  Find the volume $$E$$ of the solid. Round your answer to four decimal places.

27. Convert the integral $$\displaystyle\int_0^1 \int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} \int_{x^2+y^2}^{\sqrt{x^2+y^2}} xz \space dz \space dx \space dy$$ into an integral in cylindrical coordinates.

$$\displaystyle\int_0^1 \int_0^{\pi} \int_{r^2}^r zr^2 \space \cos \theta \, dz \space d\theta \space dr$$

28. Convert the integral $$\displaystyle \int_0^2 \int_0^y \int_0^1 (xy + z) \, dz \space dx \space dy$$ into an integral in cylindrical coordinates.

In exercises 29 - 32, evaluate the triple integral $$\displaystyle \iiint_B f(x,y,z) \,dV$$ over the solid $$B$$.

29. $$f(x,y,z) = 1, \space B = \big\{(x,y,z)\, | \,x^2 + y^2 + z^2 \leq 90, \space z \geq 0\big\}$$

[Hide Solution]

$$180 \pi \sqrt{10}$$

30. $$f(x,y,z) = 1 - \sqrt{x^2 + y^2 + z^2}, \space B = \big\{(x,y,z)\, | \,x^2 + y^2 + z^2 \leq 9, \space y \geq 0, \space z \geq 0\big\}$$

31. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B$$ is bounded above by the half-sphere $$x^2 + y^2 + z^2 = 9$$ with $$z \geq 0$$ and below by the cone $$3z^2 = x^2 + y^2$$.

$$\frac{81\pi}{4}(\frac{\pi}{3} - \frac{\sqrt{3}}{4})$$

32. $$f(x,y,z) = \sqrt{x^2 + y^2}, \space B$$ is bounded above by the half-sphere $$x^2 + y^2 + z^2 = 16$$ with $$z \geq 0$$ and below by the cone $$2z^2 = x^2 + y^2$$.

33. Show that if $$F ( \rho,\theta,\varphi) = f(\rho)g(\theta)h(\varphi)$$ is a continuous function on the spherical box $$B = \big\{(\rho,\theta,\varphi)\, | \,a \leq \rho \leq b, \space \alpha \leq \theta \leq \beta, \space \gamma \leq \varphi \leq \psi\big\}$$, then $$\displaystyle\iiint_B F \space dV = \left(\int_a^b \rho^2 f(\rho) \space dr \right) \left( \int_{\alpha}^{\beta} g (\theta) \space d\theta \right)\left( \int_{\gamma}^{\psi} h (\varphi) \space \sin \varphi \space d\varphi \right).$$

34. A function $$F$$ is said to have spherical symmetry if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as $$F(x,y,z) = f(\rho)$$, where $$\rho = \sqrt{x^2 + y^2 + z^2}$$. Show that $$\displaystyle\iiint_B F(x,y,z) \,dV = 2\pi \int_a^b \rho^2 f(\rho) \,d\rho,$$ where $$B$$ is the region between the upper concentric hemispheres of radii $$a$$ and $$b$$ centered at the origin, with $$0 < a < b$$ and $$F$$ a spherical function defined on $$B$$.

Use the previous result to show that $$\displaystyle\iiint_B (x^2 + y^2 + z^2) \sqrt{x^2 + y^2 + z^2} dV = 21 \pi,$$ where $$B = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0\big\}$$.

35. Let $$B$$ be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where $$0 < a < b$$. Consider F a function defined on B whose form in spherical coordinates $$(\rho,\theta,\varphi)$$ is $$F(x,y,z) = f(\rho)\cos \varphi$$. Show that if $$g(a) = g(b) = 0$$ and $$\displaystyle\int_a^b h (\rho) \, d\rho = 0,$$ then $$\displaystyle\iiint_B F(x,y,z)\,dV = \frac{\pi^2}{4} [ah(a) - bh(b)],$$ where $$g$$ is an antiderivative of $$f$$ and $$h$$ is an antiderivative of $$g$$.

Use the previous result to show that $$\displaystyle \iiint_B = \frac{z \cos \sqrt{x^2 + y^2 + z^2}}{\sqrt{x^2 + y^2 + z^2}} \, dV = \frac{3\pi^2}{2},$$ where $$B$$ is the region between the upper concentric hemispheres of radii $$\pi$$ and $$2\pi$$ centered at the origin and situated in the first octant.

In exercises 36 - 39, the function $$f$$ and region $$E$$ are given.

a. Express the region $$E$$ and function $$f$$ in cylindrical coordinates.

b. Convert the integral $$\displaystyle \iiint_B f(x,y,z)\, dV$$ into cylindrical coordinates and evaluate it.

36. $$f(x,y,z) = z; \space E = \big\{(x,y,z)\, | \,0 \leq x^2 + y^2 + z^2 \leq 1, \space z \geq 0\big\}$$

37. $$f(x,y,z) = x + y; \space E = \big\{(x,y,z)\, | \,1 \leq x^2 + y^2 + z^2 \leq 2, \space z \geq 0, \space y \geq 0\big\}$$

a. $$f(\rho,\theta, \varphi) = \rho \space \sin \varphi \space (\cos \theta + \sin \theta), \space E = \big\{(\rho,\theta,\varphi)\, | \,1 \leq \rho \leq 2, \space 0 \leq \theta \leq \pi, \space 0 \leq \varphi \leq \frac{\pi}{2}\big\}$$;

b. $$\displaystyle \int_0^{\pi} \int_0^{\pi/2} \int_1^2 \rho^3 \cos \varphi \space \sin \varphi \space d\rho \space d\varphi \space d\theta = \frac{15\pi}{8}$$

38. $$f(x,y,z) = 2xy; \space E = \big\{(x,y,z)\, | \,\sqrt{x^2 + y^2} \leq z \leq \sqrt{1 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\big\}$$

39. $$f(x,y,z) = z; \space E = \big\{(x,y,z)\, | \,x^2 + y^2 + z^2 - 2x \leq 0, \space \sqrt{x^2 + y^2} \leq z\big\}$$

a. $$f(\rho,\theta,\varphi) = \rho \space \cos \varphi; \space E = \big\{(\rho,\theta,\varphi)\, | \,0 \leq \rho \leq 2 \space \cos \varphi, \space 0 \leq \theta \leq \frac{\pi}{2}, \space 0 \leq \varphi \leq \frac{\pi}{4}\big\}$$;

b. $$\displaystyle\int_0^{\pi/2} \int_0^{\pi/4} \int_0^{2 \space \cos \varphi} \rho^3 \sin \varphi \space \cos \varphi \space d\rho \space d\varphi \space d\theta = \frac{7\pi}{24}$$

In exercises 40 - 41, find the volume of the solid $$E$$ whose boundaries are given in rectangular coordinates.

40. $$E = \big\{ (x,y,z)\, | \,\sqrt{x^2 + y^2} \leq z \leq \sqrt{16 - x^2 - y^2}, \space x \geq 0, \space y \geq 0\big\}$$

41. $$E = \big\{ (x,y,z)\, | \,x^2 + y^2 + z^2 - 2z \leq 0, \space \sqrt{x^2 + y^2} \leq z\big\}$$

$$\frac{\pi}{4}$$

42. Use spherical coordinates to find the volume of the solid situated outside the sphere $$\rho = 1$$ and inside the sphere $$\rho = \cos \varphi$$, with $$\varphi \in [0,\frac{\pi}{2}]$$.

43. Use spherical coordinates to find the volume of the ball $$\rho \leq 3$$ that is situated between the cones $$\varphi = \frac{\pi}{4}$$ and $$\varphi = \frac{\pi}{3}$$.

$$9\pi (\sqrt{2} - 1)$$

44. Convert the integral $$\displaystyle \int_{-4}^4 \int_{-\sqrt{16-y^2}}^{\sqrt{16-y^2}} \int_{-\sqrt{16-x^2-y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2) \, dz \, dx \, dy$$ into an integral in spherical coordinates.

45. Convert the integral $$\displaystyle \int_0^4 \int_0^{\sqrt{16-x^2}} \int_{-\sqrt{16-x^2-y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^2 \, dz \space dy \space dx$$ into an integral in spherical coordinates.

$$\displaystyle\int_0^{\pi/2} \int_0^{\pi/2} \int_0^4 \rho^6 \sin \varphi \, d\rho \, d\phi \, d\theta$$

47. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates $$\displaystyle \int_{\pi/2}^{\pi} \int_{5\pi}^{\pi/6} \int_0^2 \rho^2 \sin \varphi \space d\rho \space d\varphi \space d\theta.$$ Find the volume $$V$$ of the solid. Round your answer to three decimal places.

$$V = \frac{4\pi\sqrt{3}}{3} \approx 7.255$$

48. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as $$\displaystyle \int_0^{2\pi} \int_{3\pi/4}^{\pi/4} \int_0^1 \rho^2 \sin \varphi \space d\rho \space d\varphi \space d\theta.$$ Find the volume $$V$$ of the solid. Round your answer to three decimal places.

49. [T] Use a CAS to evaluate the integral $$\displaystyle \iiint_E (x^2 + y^2) \, dV$$ where $$E$$ lies above the paraboloid $$z = x^2 + y^2$$ and below the plane $$z = 3y$$.

$$\frac{343\pi}{32}$$

50. [T]

a. Evaluate the integral $$\displaystyle \iiint_E e^{\sqrt{x^2+y^2+z^2}}\, dV,$$ where $$E$$ is bounded by spheres $$4x^2 + 4y^2 + 4z^2 = 1$$ and $$x^2 + y^2 + z^2 = 1$$.

b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.

51. Express the volume of the solid inside the sphere $$x^2 + y^2 + z^2 = 16$$ and outside the cylinder $$x^2 + y^2 = 4$$ as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

$$\displaystyle \int_0^{2\pi}\int_2^4\int_{−\sqrt{16−r^2}}^{\sqrt{16−r^2}}r\,dz\,dr\,dθ$$  and  $$\displaystyle \int_{\pi/6}^{5\pi/6}\int_0^{2\pi}\int_{2\csc \phi}^{4}\rho^2\sin \rho \, d\rho \, d\theta \, d\phi$$

52. Express the volume of the solid inside the sphere $$x^2 + y^2 + z^2 = 16$$ and outside the cylinder $$x^2 + y^2 = 4$$ that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

53. The power emitted by an antenna has a power density per unit volume given in spherical coordinates by $$p(\rho,\theta,\varphi) = \frac{P_0}{\rho^2} \cos^2 \theta \space \sin^4 \varphi$$, where $$P_0$$ is a constant with units in watts. The total power within a sphere $$B$$ of radius $$r$$ meters is defined as $$\displaystyle P = \iiint_B p(\rho,\theta,\varphi) \, dV.$$ Find the total power $$P$$.

$$P = \frac{32P_0 \pi}{3}$$ watts

54. Use the preceding exercise to find the total power within a sphere $$B$$ of radius 5 meters when the power density per unit volume is given by $$p(\rho, \theta,\varphi) = \frac{30}{\rho^2} \cos^2 \theta \sin^4 \varphi$$.

55. A charge cloud contained in a sphere $$B$$ of radius $$r$$ centimeters centered at the origin has its charge density given by $$q(x,y,z) = k\sqrt{x^2 + y^2 + z^2}\frac{\mu C}{cm^3}$$, where $$k > 0$$.  The total charge contained in $$B$$ is given by $$\displaystyle Q = \iiint_B q(x,y,z) \, dV.$$ Find the total charge $$Q$$.

$$Q = kr^4 \pi \mu C$$

56. Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is $$q(x,y,z) = 20 \sqrt{x^2 + y^2 + z^2} \frac{\mu C}{cm^3}$$.

4.6: Calculating Centers of Mass and Moments of Inertia

In exercises 1 - 12, the region $$R$$ occupied by a lamina is shown in a graph. Find the mass of $$R$$ with the density function $$\rho$$.

1. $$R$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(6,0); \space \rho (x,y) = xy$$.

$$\frac{27}{2}$$

2. $$R$$ is the triangular region with vertices $$(0,0), \space (1,1)$$, and $$(0,5); \space \rho (x,y) = x + y$$.

3. $$R$$ is the rectangular region with vertices $$(0,0), \space (0,3), \space (6,3)$$ and $$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

$$24\sqrt{2}$$

4. $$R$$ is the rectangular region with vertices $$(0,1), \space (0,3), \space (3,3)$$ and $$(3,1); \space \rho (x,y) = x^2y$$.

5. $$R$$ is the trapezoidal region determined by the lines $$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, and $$x = 0; \space \rho (x,y) = 3xy$$.

$$76$$

6. $$R$$ is the trapezoidal region determined by the lines $$y = 0, \space y = 1, \space y = x$$ and $$y = -x + 3; \space \rho (x,y) = 2x + y$$.

7. $$R$$ is the disk of radius $$2$$ centered at $$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.

$$8\pi$$

8. $$R$$ is the unit disk; $$\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4$$.

9. $$R$$ is the region enclosed by the ellipse $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$.

$$\frac{\pi}{2}$$

10. $$R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}$$.

11. $$R$$ is the region bounded by $$y = x, \space y = -x, \space y = x + 2, \space y = -x + 2; \space \rho(x,y) = 1$$.

$$2$$

12. $$R$$ is the region bounded by $$y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1$$, and $$y = 2; \space \rho (x,y) = 4(x + y)$$.

In exercises 13 - 24, consider a lamina occupying the region $$R$$ and having the density function $$\rho$$ given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions.

a. Find the moments $$M_x$$ and $$M_y$$ about the $$x$$-axis and $$y$$-axis, respectively.

b. Calculate and plot the center of mass of the lamina.

c. [T] Use a CAS to locate the center of mass on the graph of $$R$$.

13. [T] $$R$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(6,0); \space \rho (x,y) = xy$$.

a. $$M_x = \frac{81}{5}, \space M_y = \frac{162}{5}$$;
b. $$\bar{x} = \frac{12}{5}, \space \bar{y} = \frac{6}{5}$$;
c.

14. [T] $$R$$ is the triangular region with vertices $$(0,0), \space (1,1)$$, and $$(0,5); \space \rho (x,y) = x + y$$.

15. [T] $$R$$ is the rectangular region with vertices $$(0,0), \space (0,3), \space (6,3)$$, and $$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

a. $$M_x = \frac{216\sqrt{2}}{5}, \space M_y = \frac{432\sqrt{2}}{5}$$;
b. $$\bar{x} = \frac{18}{5}, \space \bar{y} = \frac{9}{5}$$;
c.

16. [T] $$R$$ is the rectangular region with vertices $$(0,1), \space (0,3), \space (3,3)$$, and $$(3,1); \space \rho (x,y) = x^2y$$.

17. [T] $$R$$ is the trapezoidal region determined by the lines $$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, and $$x = 0; \space \rho (x,y) = 3xy$$.

a. $$M_x = \frac{368}{5}, \space M_y = \frac{1552}{5}$$;
b. $$\bar{x} = \frac{92}{95}, \space \bar{y} = \frac{388}{95}$$;
c.

18. [T] $$R$$ is the trapezoidal region determined by the lines $$y = 0, \space y = 1, \space y = x,$$ and $$y = -x + 3; \space \rho (x,y) = 2x + y$$.

19. [T] $$R$$ is the disk of radius $$2$$ centered at $$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.

a. $$M_x = 16\pi, \space M_y = 8\pi$$;
b. $$\bar{x} = 1, \space \bar{y} = 2$$;
c.

20. [T] $$R$$ is the unit disk; $$\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4$$.

21. [T] $$R$$ is the region enclosed by the ellipse $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$.

a. $$M_x = 0, \space M_y = 0)$$;
b. $$\bar{x} = 0, \space \bar{y} = 0$$;
c.

22. [T] $$R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}$$.

23. [T] $$R$$ is the region bounded by $$y = x, \space y = -x, \space y = x + 2$$, and $$y = -x + 2; \space \rho (x,y) = 1$$.

a. $$M_x = 2, \space M_y = 0)$$;
b. $$\bar{x} = 0, \space \bar{y} = 1$$;
c.

24. [T] $$R$$ is the region bounded by $$y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1$$, and $$y = 2; \space \rho (x,y) = 4(x + y)$$.

In exercises 25 - 36, consider a lamina occupying the region $$R$$ and having the density function $$\rho$$ given in the first two groups of Exercises.

a. Find the moments of inertia $$I_x, \space I_y$$ and $$I_0$$ about the $$x$$-axis, $$y$$-axis, and origin, respectively.

b. Find the radii of gyration with respect to the $$x$$-axis, $$y$$-axis, and origin, respectively.

25. $$R$$ is the triangular region with vertices $$(0,0), \space (0,3)$$, and $$(6,0); \space \rho (x,y) = xy$$.

a. $$I_x = \frac{243}{10}, \space I_y = \frac{486}{5}$$, and $$I_0 = \frac{243}{2}$$;
b. $$R_x = \frac{3\sqrt{5}}{5}, \space R_y = \frac{6\sqrt{5}}{5}$$, and $$R_0 = 3$$

26. $$R$$ is the triangular region with vertices $$(0,0), \space (1,1)$$, and $$(0,5); \space \rho (x,y) = x + y$$.

27. $$R$$ is the rectangular region with vertices $$(0,0), \space (0,3), \space (6,3)$$, and $$(6,0); \space \rho (x,y) = \sqrt{xy}$$.

a. $$I_x = \frac{2592\sqrt{2}}{7}, \space I_y = \frac{648\sqrt{2}}{7}$$, and $$I_0 = \frac{3240\sqrt{2}}{7}$$;
b. $$R_x = \frac{6\sqrt{21}}{7}, \space R_y = \frac{3\sqrt{21}}{7}$$, and $$R_0 = \frac{3\sqrt{106}}{7}$$

28. $$R$$ is the rectangular region with vertices $$(0,1), \space (0,3), \space (3,3)$$, and $$(3,1); \space \rho (x,y) = x^2y$$.

29. $$R$$ is the trapezoidal region determined by the lines $$y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2$$, and x = 0; \space \rho (x,y) = 3xy\).

a. $$I_x = 88, \space I_y = 1560$$, and $$I_0 = 1648$$;
b. $$R_x = \frac{\sqrt{418}}{19}, \space R_y = \frac{\sqrt{7410}}{10}$$, and $$R_0 = \frac{2\sqrt{1957}}{19}$$

30. $$R$$ is the trapezoidal region determined by the lines $$y = 0, \space y = 1, \space y = x$$, and y = -x + 3; \space \rho (x,y) = 2x + y\).

31. $$R$$ is the disk of radius $$2$$ centered at $$(1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5$$.

a. $$I_x = \frac{128\pi}{3}, \space I_y = \frac{56\pi}{3}$$, and $$I_0 = \frac{184\pi}{3}$$;
b. $$R_x = \frac{4\sqrt{3}}{3}, \space R_y = \frac{\sqrt{21}}{2}$$, and $$R_0 = \frac{\sqrt{69}}{3}$$

32. $$R$$ is the unit disk; $$\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4$$.

33. $$R$$ is the region enclosed by the ellipse $$x^2 + 4y^2 = 1; \space \rho(x,y) = 1$$.

a. $$I_x = \frac{\pi}{32}, \space I_y = \frac{\pi}{8}$$, and $$I_0 = \frac{5\pi}{32}$$;
b. $$R_x = \frac{1}{4}, \space R_y = \frac{1}{2}$$, and $$R_0 = \frac{\sqrt{5}}{4}$$

34. $$R = \big\{(x,y) \,|\, 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\big\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}$$.

35. $$R$$ is the region bounded by $$y = x, \space y = -x, \space y = x + 2$$, and $$y = -x + 2; \space \rho (x,y) = 1$$.

a. $$I_x = \frac{7}{3}, \space I_y = \frac{1}{3}$$, and $$I_0 = \frac{8}{3}$$;
b. $$R_x = \frac{\sqrt{42}}{6}, \space R_y = \frac{\sqrt{6}}{6}$$, and $$R_0 = \frac{2\sqrt{3}}{3}$$

36. $$R$$ is the region bounded by $$y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1$$, and $$y = 2; \space \rho (x,y) = 4(x + y)$$.

37. Let $$Q$$ be the solid unit cube. Find the mass of the solid if its density $$\rho$$ is equal to the square of the distance of an arbitrary point of $$Q$$ to the $$xy$$-plane.

$$m = \frac{1}{3}$$

38. Let $$Q$$ be the solid unit hemisphere. Find the mass of the solid if its density $$\rho$$ is proportional to the distance of an arbitrary point of $$Q$$ to the origin.

39. The solid $$Q$$ of constant density $$1$$ is situated inside the sphere $$x^2 + y^2 + z^2 = 16$$ and outside the sphere $$x^2 + y^2 + z^2 = 1$$. Show that the center of mass of the solid is not located within the solid.

40. Find the mass of the solid $$Q = \big\{ (x,y,z) \,|\, 1 \leq x^2 + z^2 \leq 25, \space y \leq 1 - x^2 - z^2 \big\}$$ whose density is $$\rho (x,y,z) = k$$, where $$k > 0$$.

41. [T] The solid $$Q = \big\{ (x,y,z) \,|\, x^2 + y^2 \leq 9, \space 0 \leq z \leq 1, \space x \geq 0, \space y \geq 0\big\}$$ has density equal to the distance to the $$xy$$-plane. Use a CAS to answer the following questions.

a. Find the mass of $$Q$$.

b. Find the moments $$M_{xy}, \space M_{xz}$$ and $$M_{yz}$$ about the $$xy$$-plane, $$xz$$-plane, and $$yz$$-plane, respectively.

c. Find the center of mass of $$Q$$.

d. Graph $$Q$$ and locate its center of mass.

a. $$m = \frac{9\pi}{4}$$;
b. $$M_{xy} = \frac{3\pi}{2}, \space M_{xz} = \frac{81}{8}, \space M_{yz} = \frac{81}{8}$$;
c. $$\bar{x} = \frac{9}{2\pi}, \space \bar{y} = \frac{9}{2\pi}, \space \bar{z} = \frac{2}{3}$$;
d.

42. Consider the solid $$Q = \big\{ (x,y,z) \,|\, 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\big\}$$ with the density function $$\rho(x,y,z) = x + y + 1$$.

a. Find the mass of $$Q$$.

b. Find the moments $$M_{xy}, \space M_{xz}$$ and $$M_{yz}$$ about the $$xy$$-plane, $$xz$$-plane, and $$yz$$-plane, respectively.

c. Find the center of mass of $$Q$$.

43. [T] The solid $$Q$$ has the mass given by the triple integral $$\displaystyle \int_{-1}^1 \int_0^{\pi/4} \int_0^1 r^2 \, dr \space d\theta \space dz.$$

Use a CAS to answer the following questions.

• Show that the center of mass of $$Q$$ is located in the $$xy$$-plane.
• Graph $$Q$$ and locate its center of mass.

$$\bar{x} = \frac{3\sqrt{2}}{2\pi}$$, $$\bar{y} = \frac{3(2-\sqrt{2})}{2\pi}, \space \bar{z} = 0$$; 2. the solid $$Q$$ and its center of mass are shown in the following figure.

44. The solid $$Q$$ is bounded by the planes $$x + 4y + z = 8, \space x = 0, \space y = 0$$, and $$z = 0$$. Its density at any point is equal to the distance to the $$xz$$-plane. Find the moments of inertia of the solid about the $$xz$$-plane.

45. The solid $$Q$$ is bounded by the planes $$x + y + z = 3, \space x = 0, \space y = 0$$, and $$z = 0$$. Its density is $$\rho(x,y,z) = x + ay$$, where $$a > 0$$. Show that the center of mass of the solid is located in the plane $$z = \frac{3}{5}$$ for any value of $$a$$.

46. Let $$Q$$ be the solid situated outside the sphere $$x^2 + y^2 + z^2 = z$$ and inside the upper hemisphere $$x^2 + y^2 + z^2 = R^2$$, where $$R > 1$$. If the density of the solid is $$\rho (x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}$$, find $$R$$ such that the mass of the solid is $$\frac{7\pi}{2}.$$

47. The mass of a solid $$Q$$ is given by $$\displaystyle \int_0^2 \int_0^{\sqrt{4-x^2}} \int_{\sqrt{x^2+y^2}}^{\sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^n \, dz \space dy \space dx,$$ where $$n$$ is an integer. Determine $$n$$ such the mass of the solid is $$(2 - \sqrt{2}) \pi$$.

$$n = -2$$

48. Let $$Q$$ be the solid bounded above the cone $$x^2 + y^2 = z^2$$ and below the sphere $$x^2 + y^2 + z^2 - 4z = 0$$. Its density is a constant $$k > 0$$. Find $$k$$ such that the center of mass of the solid is situated $$7$$ units from the origin.

49. The solid $$Q = \big\{(x,y,z) \,|\, 0 \leq x^2 + y^2 \leq 16, \space x \geq 0, \space y \geq 0, \space 0 \leq z \leq x\big\}$$ has the density $$\rho (x,y,z) = k$$. Show that the moment $$M_{xy}$$ about the $$xy$$-plane is half of the moment $$M_{yz}$$ about the $$yz$$-plane.

50. The solid $$Q$$ is bounded by the cylinder $$x^2 + y^2 = a^2$$, the paraboloid $$b^2 - z = x^2 + y^2$$, and the $$xy$$-plane, where $$0 < a < b$$. Find the mass of the solid if its density is given by $$\rho(x,y,z) = \sqrt{x^2 + y^2}$$.

51. Let $$Q$$ be a solid of constant density $$k$$, where $$k > 0$$, that is located in the first octant, inside the circular cone $$x^2 + y^2 = 9(z - 1)^2$$, and above the plane $$z = 0$$. Show that the moment $$M_{xy}$$ about the $$xy$$-plane is the same as the moment $$M_{yz}$$ about the $$xz$$-plane.

52. The solid $$Q$$ has the mass given by the triple integral $$\displaystyle \int_0^1 \int_0^{\pi/2} \int_0^{r^3} (r^4 + r) \space dz \space d\theta \space dr.$$

a. Find the density of the solid in rectangular coordinates.

b. Find the moment $$M_{xy}$$ about the $$xy$$-plane.

53. The solid $$Q$$ has the moment of inertia $$I_x$$ about the $$yz$$-plane given by the triple integral $$\displaystyle \int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{\frac{1}{2}(x^2+y^2)}^{\sqrt{x^2+y^2}} (y^2 + z^2)(x^2 + y^2) \, dz \space dx \space dy.$$

a. Find the density of $$Q$$.

b. Find the moment of inertia $$I_z$$ about the $$xy$$-plane.

a. $$\rho (x,y,z) = x^2 + y^2$$;
b. $$\frac{16\pi}{7}$$

54. The solid $$Q$$ has the mass given by the triple integral $$\displaystyle \int_0^{\pi/4} \int_0^{2 \space sec \space \theta} \int_0^1 (r^3 \cos \theta \sin \theta + 2r) \, dz \space dr \space d\theta.$$

a. Find the density of the solid in rectangular coordinates.

b. Find the moment $$M_{xz}$$ about the $$xz$$-plane.

55. Let $$Q$$ be the solid bounded by the $$xy$$-plane, the cylinder $$x^2 + y^2 = a^2$$, and the plane $$z = 1$$, where $$a > 1$$ is a real number. Find the moment $$M_{xy}$$ of the solid about the $$xy$$-plane if its density given in cylindrical coordinates is $$\rho(x,y,z) = \frac{d^2f}{dr^2} (r)$$, where $$f$$ is a differentiable function with the first and second derivatives continuous and differentiable on $$(0,a)$$.

$$M_{xy} = \pi (f(0) - f(a) + af'(a))$$

56. A solid $$Q$$ has a volume given by $$\displaystyle \iint_D \int_a^b \, dA \space dz$$, where $$D$$ is the projection of the solid onto the $$xy$$-plane and $$a < b$$ are real numbers, and its density does not depend on the variable $$z$$. Show that its center of mass lies in the plane $$z = \frac{a+b}{2}$$.

57. Consider the solid enclosed by the cylinder $$x^2 + z^2 = a^2$$ and the planes $$y = b$$ and $$y = c$$, where $$a > 0$$ and $$b < c$$ are real numbers. The density of $$Q$$ is given by $$\rho(x,y,z) = f'(y)$$, where $$f$$ is a differential function whose derivative is continuous on $$(b,c)$$. Show that if $$f(b) = f(c)$$, then the moment of inertia about the $$xz$$-plane of $$Q$$ is null.

58. [T] The average density of a solid $$Q$$ is defined as $$\displaystyle \rho_{ave} = \frac{1}{V(Q)} \iiint_Q \rho(x,y,z) \, dV = \frac{m}{V(Q)},$$ where $$V(Q)$$ and $$m$$ are the volume and the mass of $$Q$$, respectively. If the density of the unit ball centered at the origin is $$\rho (x,y,z) = e^{-x^2-y^2-z^2}$$, use a CAS to find its average density. Round your answer to three decimal places.

59. Show that the moments of inertia $$I_x, \space I_y$$, and $$I_z$$ about the $$yz$$-plane, $$xz$$-plane, and $$xy$$-plane, respectively, of the unit ball centered at the origin whose density is $$\rho (x,y,z) = e^{-x^2-y^2-z^2}$$ are the same. Round your answer to two decimal places.

$$I_x = I_y = I_z \approx 0.84$$

4.7: Change of Variables in Multiple Integrals

In exercises 1 - 6, the function $$T : S \rightarrow R, \space T (u,v) = (x,y)$$ on the region $$S = \big\{(u,v) \,|\, 0 \leq u \leq 1, \space 0 \leq v \leq 1\big\}$$ bounded by the unit square is given, where $$R \in R^2$$ is the image of $$S$$ under $$T$$.

a. Justify that the function $$T$$ is a $$C^1$$ transformation.

b. Find the images of the vertices of the unit square $$S$$ through the function $$T$$.

c. Determine the image $$R$$ of the unit square $$S$$ and graph it.

1. $$x = 2u, \space y = 3v$$

2. $$x = \frac{u}{2}, \space y = \frac{v}{3}$$

a. $$T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = \frac{u}{2}$$ and $$y = h(u,v) = \frac{v}{3}$$. The functions $$g$$ and $$h$$ are continuous and differentiable, and the partial derivatives $$g_u (u,v) = \frac{1}{2}, \space g_v (u,v) = 0, \space h_u (u,v) = 0$$ and $$h_v (u,v) = \frac{1}{3}$$ are continuous on $$S$$;

b. $$T(0,0) = (0,0), \space T(1,0) = \left(\frac{1}{2},0\right), \space T(0,1) = \left(0,\frac{1}{3}\right)$$, and $$T(1,1) = \left(\frac{1}{2}, \frac{1}{3} \right)$$;

c. $$R$$ is the rectangle of vertices $$(0,0), \space \left(0,\frac{1}{3}\right), \space \left(\frac{1}{2}, \frac{1}{3} \right)$$, and $$\left(0,\frac{1}{3}\right)$$ in the $$xy$$-plane; the following figure.

3. $$x = u - v, \space y = u + v$$

4. $$x = 2u - v, \space y = u + 2v$$

a. $$T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = 2u - v$$ and $$y = h(u,v) = u + 2v$$. The functions $$g$$ and $$h$$ are continuous and differentiable, and the partial derivatives $$g_u (u,v) = 2, \space g_v (u,v) = -1, \space h_u (u,v) = 1$$ and $$h_v (u,v) = 2$$ are continuous on $$S$$;

b. $$T(0,0) = (0,0), \space T(1,0) = (2,1), \space T(0,1) = (-1,2)$$, and $$T(1,1) = (1,3)$$;

c. $$R$$ is the parallelogram of vertices $$(0,0), \space (2,1) \space (1,3)$$, and $$(-1,2)$$ in the $$xy$$-plane; the following figure.

5. $$x = u^2, \space y = v^2$$

6. $$x = u^3, \space y = v^3$$

a. $$T(u,v) = (g(u,v), \space h(u,v), \space x = g(u,v) = u^3$$ and $$y = h(u,v) = v^3$$. The functions $$g$$ and $$h$$ are continuous and differentiable, and the partial derivatives $$g_u (u,v) = 3u^2, \space g_v (u,v) = 0, \space h_u (u,v) = 0$$ and $$h_v (u,v) = 3v^2$$ are continuous on $$S$$;

b. $$T(0,0) = (0,0), \space T(1,0) = (1,0), \space T(0,1) = (0,1)$$, and $$T(1,1) = (1,1)$$;

c. $$R$$ is the unit square in the $$xy$$-plane see the figure in the answer to the previous exercise.

In exercises 7 - 12, determine whether the transformations $$T : S \rightarrow R$$ are one-to-one or not.

7. $$x = u^2, \space y = v^2$$, where $$S$$ is the rectangle of vertices $$(-1,0), \space (1,0), \space (1,1)$$, and $$(-1,1)$$.

8. $$x = u^4, \space y = u^2 + v$$, where $$S$$ is the triangle of vertices $$(-2,0), \space (2,0)$$, and $$(0,2)$$.

$$T$$ is not one-to-one: two points of $$S$$ have the same image. Indeed, $$T(-2,0) = T(2,0) = (16,4)$$.

9. $$x = 2u, \space y = 3v$$, where $$S$$ is the square of vertices $$(-1,1), \space (-1,-1), \space (1,-1)$$, and $$(1,1)$$.

10. $$T(u, v) = (2u - v, u),$$ where $$S$$ is the triangle with vertices $$(-1,1), \, (-1,-1)$$, and $$(1,-1)$$.

$$T$$ is one-to-one: We argue by contradiction. $$T(u_1,v_1) = T(u_2,v_2)$$ implies $$2u_1 - v_1 = 2u_2 - v_2$$ and $$u_1 = u_2$$. Thus, $$u_1 = u+2$$ and $$v_1 = v_2$$.

11. $$x = u + v + w, \space y = u + v, \space z = w$$, where $$S = R = R^3$$.

12. $$x = u^2 + v + w, \space y = u^2 + v, \space z = w$$, where $$S = R = R^3$$.

$$T$$ is not one-to-one: $$T(1,v,w) = (-1,v,w)$$

In exercises 13 - 18, the transformations $$T : R \rightarrow S$$ are one-to-one. Find their related inverse transformations $$T^{-1} : R \rightarrow S$$.

13. $$x = 4u, \space y = 5v$$, where $$S = R = R^2$$.

14. $$x = u + 2v, \space y = -u + v$$, where $$S = R = R^2$$.

$$u = \frac{x-2y}{3}, \space v= \frac{x+y}{3}$$

15. $$x = e^{2u+v}, \space y = e^{u-v}$$, where $$S = R^2$$ and $$R = \big\{(x,y) \,|\, x > 0, \space y > 0\big\}$$

16. $$x = \ln u, \space y = \ln(uv)$$, where $$S = \big\{(u,v) \,|\, u > 0, \space v > 0\big\}$$ and $$R = R^2$$.

$$u = e^x, \space v = e^{-x+y}$$

17. $$x = u + v + w, \space y = 3v, \space z = 2w$$, where $$S = R = R^3$$.

18. $$x = u + v, \space y = v + w, \space z = u + w$$, where $$S = R = R^3$$.

$$u = \frac{x-y+z}{2}, \space v = \frac{x+y-z}{2}, \space w = \frac{-x+y+z}{2}$$

In exercises 19 - 22, the transformation $$T : S \rightarrow R, \space T (u,v) = (x,y)$$ and the region $$R \subset R^2$$ are given. Find the region $$S \subset R^2$$.

19. $$x = au, \space y = bv, \space R = \big\{(x,y) \,|\, x^2 + y^2 \leq a^2 b^2\big\}$$ where $$a,b > 0$$

20. $$x = au, \space y = bc, \space R = \big\{(x,y) \,|\, \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1\big\}$$, where $$a,b > 0$$

$$S = \big\{(u,v) \,|\, u^2 + v^2 \leq 1\big\}$$

21. $$x = \frac{u}{a}, \space y = \frac{v}{b}, \space z = \frac{w}{c}, \space R = \big\{(x,y)\,|\,x^2 + y^2 + z^2 \leq 1\big\}$$, where $$a,b,c > 0$$

22. $$x = au, \space y = bv, \space z = cw, \space R = \big\{(x,y)\,|\,\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} \leq 1, \space z > 0\big\}$$, where $$a,b,c > 0$$

$$R = \big\{(u,v,w)\,|\,u^2 - v^2 - w^2 \leq 1, \space w > 0\big\}$$

In exercises 23 - 32, find the Jacobian $$J$$ of the transformation.

23. $$x = u + 2v, \space y = -u + v$$

24. $$x = \frac{u^3}{2}, \space y = \frac{v}{u^2}$$

$$\frac{3}{2}$$

25. $$x = e^{2u-v}, \space y = e^{u+v}$$

26. $$x = ue^v, \space y = e^{-v}$$

$$-1$$

27. $$x = u \space \cos (e^v), \space y = u \space \sin(e^v)$$

28. $$x = v \space \sin (u^2), \space y = v \space \cos(u^2)$$

$$2uv$$

29. $$x = u \space \cosh v, \space y = u \space \sinh v, \space z = w$$

30. $$x = v \space \cosh \left(\frac{1}{u}\right), \space y = v \space \sinh \left(\frac{1}{u}\right), \space z = u + w^2$$

$$\frac{v}{u^2}$$

31. $$x = u + v, \space y = v + w, \space z = u$$

32. $$x = u - v, \space y = u + v, \space z = u + v + w$$

$$2$$

33. The triangular region $$R$$ with the vertices $$(0,0), \space (1,1)$$, and $$(1,2)$$ is shown in the following figure.

a. Find a transformation $$T : S \rightarrow R, \space T(u,v) = (x,y) = (au + bv + dv)$$, where $$a,b,c$$, and $$d$$ are real numbers with $$ad - bc \neq 0$$ such that $$T^{-1} (0,0) = (0,0), \space T^{-1} (1,1) = (1,0)$$, and $$T^{-1}(1,2) = (0,1)$$.

b. Use the transformation $$T$$ to find the area $$A(R)$$ of the region $$R$$.

34. The triangular region $$R$$ with the vertices $$(0,0), \space (2,0)$$, and $$(1,3)$$ is shown in the following figure.

a. Find a transformation $$T : S \rightarrow R, \space T(u,v) = (x,y) = (au + bv + dv)$$, where $$a,b,c$$, and $$d$$ are real numbers with $$ad - bc \neq 0$$ such that $$T^{-1} (0,0) = (0,0), \space T^{-1} (2,0) = (1,0)$$, and $$T^{-1}(1,3) = (0,1)$$.

b. Use the transformation $$T$$ to find the area $$A(R)$$ of the region $$R$$.

a. $$T (u,v) = (2u + v, \space 3v)$$
b. The area of $$R$$ is $$\displaystyle A(R) = \int_0^3 \int_{y/3}^{(6-y)/3} \, dx \, dy = \int_0^1 \int_0^{1-u} \left|\frac{\partial (x,y)}{\partial (u,v)}\right| \, dv \space du = \int_0^1 \int_0^{1-u} 6 \, dv \, du = 3.$$

In exercises 35 - 36, use the transformation $$u = y - x, \space v = y$$, to evaluate the integrals on the parallelogram $$R$$ of vertices $$(0,0), \space (1,0), \space (2,1)$$, and $$(1,1)$$ shown in the following figure.

35. $$\displaystyle \iint_R (y - x) \, dA$$

36. $$\displaystyle \iint_R (y^2 - xy) \, dA$$

$$-\frac{1}{4}$$

In exercises 37 - 38, use a change of variables to evaluate the double integrals over the given region R. Graph the region.

37. $$\displaystyle \iint_R e^{xy} \, dA$$ where $$R$$ is bounded by $$y=4x, \space y=x, \space y=4/x \space and \space y=1/x$$

$$(e^4-e)\ln 2$$

38. $$\displaystyle \iint_R \sin (x - y) \, dA$$ where $$R$$ is the square with the vertices $$(0,0), \space (1,1)$$, $$(2,0), \space and (1,-1)$$

$$1-\cos 2$$

In exercises 39 - 40, use the transformation $$x = u, \space 5y = v$$ to evaluate the integrals on the region $$R$$ bounded by the ellipse $$x^2 + 25y^2 = 1$$ shown in the following figure.

39. $$\displaystyle \iint_R \sqrt{x^2 + 25y^2} \, dA$$

40. $$\displaystyle \iint_R (x^2 + 25y^2)^2 \, dA$$

$$\frac{\pi}{15}$$

In exercises 41 - 42, use the transformation $$u = x + y, \space v = x - y$$ to evaluate the integrals on the trapezoidal region $$R$$ determined by the points $$(1,0), \space (2,0), \space (0,2)$$, and $$(0,1)$$ shown in the following figure.

41. $$\displaystyle \iint_R (x^2 - 2xy + y^2) \space e^{x+y} \, dA$$

42. $$\displaystyle \iint_R (x^3 + 3x^2y + 3xy^2 + y^3) \, dA$$

$$\frac{31}{5}$$

43. The circular annulus sector $$R$$ bounded by the circles $$4x^2 + 4y^2 = 1$$ and $$9x^2 + 9y^2 = 64$$, the line $$x = y \sqrt{3}$$, and the $$y$$-axis is shown in the following figure. Find a transformation $$T$$ from a rectangular region $$S$$ in the $$r\theta$$-plane to the region $$R$$ in the $$xy$$-plane. Graph $$S$$.

44. The solid $$R$$ bounded by the circular cylinder $$x^2 + y^2 = 9$$ and the planes $$z = 0, \space z = 1, \space x = 0$$, and $$y = 0$$ is shown in the following figure. Find a transformation $$T$$ from a cylindrical box $$S$$ in $$r\theta z$$-space to the solid $$R$$ in $$xyz$$-space.

$$T (r,\theta,z) = (r \space \cos \theta, \space r \space \sin \theta, \space z); \space S = [0,3] \times [0,\frac{\pi}{2}] \times [0,1]$$ in the $$r\theta z$$-space

45. Show that $\iint_R f \left(\sqrt{\frac{x^2}{3} + \frac{y^2}{3}}\right) dA = 2 \pi \sqrt{15} \int_0^1 f (\rho) \rho \space d\rho,$ where $$f$$ is a continuous function on $$[0,1]$$ and $$R$$ is the region bounded by the ellipse $$5x^2 + 3y^2 = 15$$.

46. Show that $\iiint_R f \left(\sqrt{16x^2 + 4y^2 + z^2}\right) dV = \frac{\pi}{2} \int_0^1 f (\rho) \rho^2 d\rho,$ where $$f$$ is a continuous function on $$[0,1]$$ and $$R$$ is the region bounded by the ellipsoid $$16x^2 + 4y^2 + z^2 = 1$$.

47. [T] Find the area of the region bounded by the curves $$xy = 1, \space xy = 3, \space y = 2x$$, and $$y = 3x$$ by using the transformation $$u = xy$$ and $$v = \frac{y}{x}$$. Use a computer algebra system (CAS) to graph the boundary curves of the region $$R$$.

48. [T] Find the area of the region bounded by the curves $$x^2y = 2, \space x^2y = 3, \space y = x$$, and $$y = 2x$$ by using the transformation $$u = x^2y$$ and $$v = \frac{y}{x}$$. Use a CAS to graph the boundary curves of the region $$R$$.

The area of $$R$$ is $$10 - 4\sqrt{6}$$; the boundary curves of $$R$$ are graphed in the following figure.

49. Evaluate the triple integral $\int_0^1 \int_1^2 \int_z^{z+1} (y + 1) \space dx \space dy \space dz$ by using the transformation $$u = x - z, \space v = 3y$$, and $$w = \frac{z}{2}$$.

50. Evaluate the triple integral $\int_0^2 \int_4^6 \int_{3z}^{3z+2} (5 - 4y) \space dx \space dy \space dz$ by using the transformation $$u = x - 3z, \space v = 4y$$, and $$w = z$$.

$$8$$

51. A transformation $$T : R^2 \rightarrow R^2, \space T (u,v) = (x,y)$$ of the form $$x = au + bv, \space y = cu + dv$$, where $$a,b,c$$, and $$d$$ are real numbers, is called linear. Show that a linear transformation for which $$ad - bc \neq 0$$ maps parallelograms to parallelograms.

52. A transformation $$T_{\theta} : R^2 \rightarrow R^2, \space T_{\theta} (u,v) = (x,y)$$ of the form $$x = u \space \cos \theta - v \space \sin \theta, \space y = u \space \sin \theta + v \space \cos \theta$$, is called a rotation angle $$\theta$$. Show that the inverse transformation of $$T_{\theta}$$ satisfies $$T_{\theta}^{-1} = T_{-\theta}$$ where $$T_{-\theta}$$ is the rotation of angle $$-\theta$$.

53. [T] Find the region $$S$$ in the $$uv$$-plane whose image through a rotation of angle $$\frac{\pi}{4}$$ is the region $$R$$ enclosed by the ellipse $$x^2 + 4y^2 = 1$$. Use a CAS to answer the following questions.

a. Graph the region $$S$$.

b. Evaluate the integral $$\displaystyle \iint_S e^{-2uv} \, du \, dv.$$ Round your answer to two decimal places.

54. [T] The transformations $$T_i : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \space i = 1, . . . , 4,$$ defined by $$T_1(u,v) = (u,-v), \space T_2 (u,v) = (-u,v), \space T_3 (u,v) = (-u, -v)$$, and $$T_4 (u,v) = (v,u)$$ are called reflections about the $$x$$-axis, $$y$$-axis origin, and the line $$y = x$$, respectively.

a. Find the image of the region $$S = \big\{(u,v)\,|\,u^2 + v^2 - 2u - 4v + 1 \leq 0\big\}$$ in the $$xy$$-plane through the transformation $$T_1 \circ T_2 \circ T_3 \circ T_4$$.

b. Use a CAS to graph $$R$$.

c. Evaluate the integral $$\displaystyle \iint_S \sin (u^2) \, du \, dv$$ by using a CAS. Round your answer to two decimal places.

a. $$R = \big\{(x,y)\,|\,y^2 + x^2 - 2y - 4x + 1 \leq 0\big\}$$;
b. $$R$$ is graphed in the following figure;

c. $$3.16$$

55. [T] The transformations $$T_{k,1,1} : \mathbb{R}^3 \rightarrow \mathbb{R}^3, \space T_{k,1,1}(u,v,w) = (x,y,z)$$ of the form $$x = ku, \space y = v, \space z = w$$, where $$k \neq 1$$ is a positive real number, is called a stretch if $$k > 1$$ and a compression if $$0 < k < 1$$ in the $$x$$-direction. Use a CAS to evaluate the integral $$\displaystyle \iiint_S e^{-(4x^2+9y^2+25z^2)} \, dx \, dy \, dz$$ on the solid $$S = \big\{(x,y,z) \,|\, 4x^2 + 9y^2 + 25z^2 \leq 1\big\}$$ by considering the compression $$T_{2,3,5}(u,v,w) = (x,y,z)$$ defined by $$x = \frac{u}{2}, \space y = \frac{v}{3}$$, and $$z = \frac{w}{5}$$. Round your answer to four decimal places.

56. [T] The transformation $$T_{a,0} : \mathbb{R}^2 \rightarrow \mathbb{R}^2, \space T_{a,0} (u,v) = (u + av, v)$$, where $$a \neq 0$$ is a real number, is called a shear in the $$x$$-direction. The transformation, $$T_{b,0} : R^2 \rightarrow R^2, \space T_{o,b}(u,v) = (u,bu + v)$$, where $$b \neq 0$$ is a real number, is called a shear in the $$y$$-direction.

a. Find transformations $$T_{0,2} \circ T_{3,0}$$.

b. Find the image $$R$$ of the trapezoidal region $$S$$ bounded by $$u = 0, \space v = 0, \space v = 1$$, and $$v = 2 - u$$ through the transformation $$T_{0,2} \circ T_{3,0}$$.

c. Use a CAS to graph the image $$R$$ in the $$xy$$-plane.

d. Find the area of the region $$R$$ by using the area of region $$S$$.

a. $$T_{0,2} \circ T_{3,0}(u,v) = (u + 3v, 2u + 7v)$$;

b. The image $$S$$ is the quadrilateral of vertices $$(0,0), \space (3,7), \space (2,4)$$, and $$(4,9)$$;

c. $$S$$ is graphed in the following figure;

d. $$\frac{3}{2}$$

57. Use the transformation, $$x = au, \space y = av, \space z = cw$$ and spherical coordinates to show that the volume of a region bounded by the spheroid $$\frac{x^2+y^2}{a^2} + \frac{z^2}{c^2} = 1$$ is $$\frac{4\pi a^2c}{3}$$.

58. Find the volume of a football whose shape is a spheroid $$\frac{x^2+y^2}{a^2} + \frac{z^2}{c^2} = 1$$ whose length from tip to tip is $$11$$ inches and circumference at the center is $$22$$ inches. Round your answer to two decimal places.

$$\frac{2662}{3\pi} \approx 282.45 \space in^3$$

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