# More Chapter 14 Review Exercises

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## 13.1: Iterated Integrals and Area

### Terms and Concepts

1. When integrating $$f_x(x,y)$$ with respect to $$x$$, the constant of integration $$C$$ is really which: $$C(x)\text{ or }C(y)$$? What does this mean?

2. Integrating an integral is called _________ __________.

3. When evaluating an iterated integral, we integrate from _______ to ________, then from _________ to __________.

4. One understanding of an iterated integral is that $$\displaystyle \int_a^b \int_{g_1(x)}^{g_2(x)}\,dy\,dx$$ gives the _______ of a plane region.

### Problems

In Exercises 5-10, evaluate the integral and subsequent iterated integral.

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

6.
(a) $$\displaystyle \int_0^\pi (2x\cos y +\sin x)\,dx$$
(b) $$\displaystyle \int_{0}^{\pi/2} \int_0^\pi (2x\cos y +\sin x)\,dx\,dy$$

7.
(a) $$\displaystyle \int_1^x (x^2y-y+2)\,dy$$
(b) $$\displaystyle \int_0^2 \int_1^x (x^2y-y+2)\,dy\,dx$$

8.
(a) $$\displaystyle \int_y^{y^2} (x-y)\,dx$$
(b) $$\displaystyle \int_{-1}^1 \int_y^{y^2} (x-y)\,dx\,dy$$

9.
(a) $$\displaystyle \int_0^{y} (\cos x \sin y)\,dx$$
(b) $$\displaystyle \int_0^\pi \int_0^{y} (\cos x \sin y)\,dx\,dy$$

10.
(a) $$\displaystyle \int_0^{x} \left (\frac{1}{1+x^2}\right )\,dy$$
(b) $$\displaystyle \int_1^2 \int_0^{x} \left (\frac{1}{1+x^2}\right )\,dy\,dx$$

In Exercises 11-16, 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.

11.

12.

13.

14.

15.

16.

In Exercises 17-22, 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.

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

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

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

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

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

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

## 13.2: Double Integration and Volume

### Terms and Concepts

1. An integral can be interpreted as giving the signed area over an interval; a double integral can be interpreted as giving the signed ________ over a region.

2. Explain why the following statement is false: "Fubini's Theorem states that $$\displaystyle \int_a^b \int_{g_1(x)}^{g_2(x)}f(x,y)\,dy\,dx = \int_a^b \int_{g_1(y)}^{g_2(y)}f(x,y)\,dx\,dy$$."

3. Explain why if $$f(x,y)>0$$ over a region $$R$$, then $$\displaystyle \iint_R f(x,y)\,dA >0$$.

4. If $$\displaystyle \iint_R f(x,y)\,dA= \displaystyle \iint_R g(x,y)\,dA$$, does this imply $$f(x,y)=g(x,y)$$?

### Problems

In Exercises 5-10,
(a) Evaluate the given iterated integral, and
(b) rewrite the integral using the other order of integration.

5. $$\displaystyle \int_1^2 \int_{-1}^1 \left ( \frac{x}{y}+3\right )\,dx\,dy$$

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

7. $$\displaystyle \int_0^4 \int_{0}^{-x/2+2} \left ( 3x^2-y+2\right )\,dy\,dx$$

8. $$\displaystyle \int_1^3 \int_{y}^3 \left ( x^2y-xy^2\right )\,dx\,dy$$

9. $$\displaystyle \int_0^21\int_{-\sqrt{1-y}}^{\sqrt{1-y}}( x+y+2 )\,dx\,dy$$

10. $$\displaystyle \int_0^9 \int_{y/3}^{\sqrt{3}} \left ( xy^2\right )\,dx\,dy$$

In Exercises 11-18:
(a) Sketch the region
$$R$$ given by the problem.
(b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region
$$R$$.
(c) Evaluate one of the iterated integrals to find the signed volume under the surface
$$z=f(x,y)$$ over the region $$R$$.

11. $$\displaystyle \iint_R x^2y\,dA$$, where $$R$$ is bounded by $$y=\sqrt{x}\text{ and }y=x^2$$.

12. $$\displaystyle \iint_R x^2y\,dA$$, where $$R$$ is bounded by $$y=\sqrt[3]{x}\text{ and }y=x^3$$.

13. $$\displaystyle \iint_R x^2-y^2\,dA$$, where $$R$$ is the rectangle with corners $$(-1,-1),(1,-1),(1,1)\text{ and }(-1,1)$$.

14. $$\displaystyle \iint_R ye^x\,dA$$, where $$R$$ is bounded by $$x=0,\,x=y^2\text{ and }y=1$$.

15. $$\displaystyle \iint_R (6-3x-2y)\,dA$$, where $$R$$ is bounded by $$x=0,y=0\text{ and }3x+2y=6$$.

16. $$\displaystyle \iint_R e^y\,dA$$, where $$R$$ is bounded by $$y=\ln x \text{ and }y=\frac{1}{e-1}(x-1)$$.

17. $$\displaystyle \iint_R (x^3y-x)\,dA$$, where $$R$$ is the half of the circle $$x^2+y^2=9$$ in the first and second quadrants.

18. $$\displaystyle \iint_R (4-sy)\,dA$$, where $$R$$ is bounded by $$y=0,y=x/e\text{ and }y=\ln x$$.

In Exercises 19-22, state why it is difficult/impossible to integrate the iterated integral in the given order of integration. Change the order of integration and evaluate the new iterated integral.

19. $$\displaystyle \int_0^4 \int_{y/2}^2 e^{x^2}\,dx\,dy$$

20. $$\displaystyle \int_0^{\sqrt{\pi/2}} \int_{x}^{\sqrt{\pi/2}} \cos (y^2)\,dy\,dx$$

21. $$\displaystyle \int_0^1 \int_{y}^1 \frac{2y}{x^2+y^2}\,dx\,dy$$

22. $$\displaystyle \int_{-1}^1 \int_{1}^2 \frac{x\tan^2 y}{1+\ln y}\,dy\,dx$$

In Exercises 23-26, find the average value of $$f$$ over the region $$R$$. Notice how these functions and regions are related to the iterated integrals given in Exercises 5-8.

23. $$f(x,y)=\frac{x}{y}+3$$; $$R$$ is the rectangle with opposite corners $$(-1,1)\text{ and }(1,2)$$.

24. $$f(x,y)=\sin x \cos y$$; $$R$$ is bounded by $$x=0,x=\pi,y=-\pi/2\text{ and }y=\pi/2$$.

25. $$f(x,y)=3x^2-y+2$$; $$R$$ is bounded by the lines $$y=0,y=2-x/2\text{ and }x=0$$.

26. $$f(x,y)=x^2y-xy^2$$; $$R$$ is bounded by $$y=x,y=1\text{ and }x=3$$.

## 13.3: Double Integration with Polar Coordinates

### Terms and Concepts

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

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

### Problems

In Exercises 3-10, a function $$f(x,y)$$ is given and a region $$R$$ of the $$xy$$-plane is described. Set up and evaluate $$\displaystyle \iint_R f(x,y)\,dA$$.

3. $$f(x,y)=3x-y+4$$; $$R$$ is the region enclosed by the circle $$x^2+y^2=1$$.

4. $$f(x,y)=4x+4y$$; $$R$$ is the region enclosed by the circle $$x^2+y^2=4$$.

5. $$f(x,y)=8-y$$; $$R$$ is the region enclosed by the circles with polar equations $$r=\cos \theta \text{ and }r=3\cos \theta$$.

6. $$f(x,y)=4$$; $$R$$ is the region enclosed by the petal of the rose curve $$r=\sin (2\theta)$$ in the first quadrant.

7. $$f(x,y)=\ln (x^2+y^2)$$; $$R$$ is the annulus enclosed by the circles $$x^2+y^2=1\text{ and }x^2+y^2=4. 8. \(f(x,y)=1-x^2-y^2$$; $$R$$ is the region enclosed by the circle $$x^2+y^2=1$$.

9. $$f(x,y)=x^2-y^2$$; $$R$$ is the region enclosed by the circle $$x^2+y^2=36$$ in the first and fourth quadrants.

10. $$f(x,y)=(x-y)/(x+y)$$; $$R$$ is the region enclosed by the lines $$y=x,y=0$$ and the circle $$x^2+y^2=1$$ in the first quadrant.

In Exercises 11-14, an iterated integral in rectangular coordinates is given. Rewrite the integral using polar coordinates and evaluate the new double integral.

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

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

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

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

In Exercises 15-16, special double integrals are presented that are especially well suited for evaluation in polar coordinates.

15. Consider $$\displaystyle \iint_R e^{-(x^2+y^2)}dA.$$
(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?

16. 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 $$\displaystyle \iint_R f(x,y)\,dA$$ in polar coordinates.

## 13.4: Center of Mass

### Terms and Concepts

1. Why is it easy to use "mass" and "weight" interchangeably, even though they are different measures?

2. Given a point $$(x,y)$$, the value of x is a measure of distance from the _________-axis.

3. We can think of $$\displaystyle \iint_R dm$$ as meaning "sum up lots of ________."

4. What is a "discrete planar system?"

5. Why does $$M_x$$ use $$\displaystyle \iint_R y\delta (x,y)\,dA$$ instead of $$\displaystyle \iint_R x\delta (x,y)\,dA$$; that is, why do we use "y" and not "x"?

6. Describe a situation where the center of mass of a lamina does not lie within the region of the lamina itself.

### Problems

In Exercises 7-10, point masses are given along a line or in the plane. Find the center of mass $$\overline{x}$$ or $$(\overline{x},\overline{y})$$, as appropriate. (All masses are in grams and distances are in cm.)

7. $$m_1 =4 \text{ at }x=1;\quad m_2=3\text{ at }x=3;\quad m_3 = 5\text{ at }x=10$$

8. $$m_1 =2 \text{ at }x=-3;\quad m_2=2\text{ at }x=-1;\quad m_3 = 3\text{ at }x=0;\quad m_4=3 \text{ at }x=7$$

9. $$m_1 =2 \text{ at }(-2,2);\quad m_2=2\text{ at }(2,-2);\quad m_3 = 20\text{ at }(0,4)$$

10. $$m_1 =1 \text{ at }(-1,1);\quad m_2=2\text{ at }(-1,1);\quad m_3 = 2\text{ at }(1,1);\quad m_4 =1\text{ at }(1,-1)$$

In Exercises 11-18, find the mass/weight of the lamina described by the region $$R$$ in the plane and its density function $$\delta (x,y)$$.

11. $$R$$ is the rectangle with corners $$(1,-3),(1,2),(7,2)\text{ and }(7,-3);\delta (x,y)=5$$gm/cm$$^2$$

12. $$R$$ is the rectangle with corners $$(1,-3),(1,2),(7,2)\text{ and }(7,-3);\delta (x,y)=(x+y^2)$$gm/cm$$^2$$

13. $$R$$ is the triangle with corners $$(-1,0),(1,0),\text{ and }(0,1);\delta (x,y)=2$$lb/in$$^2$$

14. $$R$$ is the triangle with corners $$(0,0),(1,0),\text{ and }(0,1);\delta (x,y)=(x^2+y^2+1)$$lb/in$$^2$$

15. $$R$$ is the circle centered at the origin with radius 2; $$\delta (x,y)=(x+y+4)$$kg/m$$^2$$

16. $$R$$ is the circle sector bounded by $$x^2+y^2=25$$ in the first quadrant; $$\delta (x,y) =(\sqrt{x^2+y^2}+1)$$kg/m$$^2$$

17. $$R$$ is the annulus in the first and second quadrants bounded by $$x^2+y^2=9\text{ and }x^2+y^2=36;\delta (x,y)=4$$lb/ft$$^2$$

18. $$R$$ is the annulus in the first and second quadrants bounded by $$x^2+y^2=9\text{ and }x^+y^2=36;\delta (x,y)=\sqrt{x^2+y^2}$$lb/ft$$^2$$

In Exercises 19-26, find the center of mass of the lamina described by the region $$R$$ in the plane and its density function $$\delta (x,y)$$.

Note: these are the same lamina as in Exercises 11-18.

19. $$R$$ is the rectangle with corners $$(1,-3),(1,2),(7,2)\text{ and }(7,-3);\delta (x,y)=5$$gm/cm$$^2$$

20. $$R$$ is the rectangle with corners $$(1,-3),(1,2),(7,2)\text{ and }(7,-3);\delta (x,y)=(x+y^2)$$gm/cm$$^2$$

21. $$R$$ is the triangle with corners $$(-1,0),(1,0),\text{ and }(0,1);\delta (x,y)=2$$lb/in$$^2$$

22. $$R$$ is the triangle with corners $$(0,0),(1,0),\text{ and }(0,1);\delta (x,y)=(x^2+y^2+1)$$lb/in$$^2$$

23. $$R$$ is the circle centered at the origin with radius 2; $$\delta (x,y)=(x+y+4)$$kg/m$$^2$$

24. $$R$$ is the circle sector bounded by $$x^2+y^2=25$$ in the first quadrant; $$\delta (x,y) =(\sqrt{x^2+y^2}+1)$$kg/m$$^2$$

25. $$R$$ is the annulus in the first and second quadrants bounded by $$x^2+y^2=9\text{ and }x^2+y^2=36;\delta (x,y)=4$$lb/ft$$^2$$

26. $$R$$ is the annulus in the first and second quadrants bounded by $$x^2+y^2=9\text{ and }x^+y^2=36;\delta (x,y)=\sqrt{x^2+y^2}$$lb/ft$$^2$$

The moment of inertia $$i$$ is a measure of the tendency of lamina to resist rotating about an axis or continue to rotate about an axis. $$i_x$$ is the moment of inertia about the $$x$$-axis, $$i_x$$ is the moment of inertia about the $$x$$-axis, and $$i_o$$ is the moment of inertia about the origin. These are computed as follows:

• $$i_x = \displaystyle \iint_R y^2\,dm$$
• $$i_y = \displaystyle \iint_R x^2\,dm$$
• $$i_o = \displaystyle \iint_R (x^2+y^2)\,dm$$

In Exercises 27-30, a lamina corresponding to a planar region $$R$$ is given with a mass of 16 units. For each, compute $$i_x$$, $$i_y$$ and $$i_o$$.

27. $$R$$ is the 4 x 4 square with corners $$(-2,-2) \text{ and }(2,2)$$ with density $$\delta (x,y)=1$$.

28. $$R$$ is the 8 x 2 rectangle with corners $$(-4,-1) \text{ and }(4,1)$$ with density $$\delta (x,y)=1$$.

29. $$R$$ is the 4 x 2 rectangle with corners $$(-2,-1) \text{ and }(2,1)$$ with density $$\delta (x,y)=2$$.

30. $$R$$ is the circle with radius 2 centered at the origin with density $$\delta (x,y)=4/\pi$$.

## 13.5: Surface Area

### Terms and Concepts

1. "Surface area" is analogous to what previously studied concept?

2. To approximate the area of a small portion of a surface, we computed the area of its ______ plane.

3. We interpret $$\displaystyle \iint_R \,dS$$ as "sum up lots of little _______ ________."

4. Why is it important to know how to set up a double integral to compute surface area, even if the resulting integral is hard to evaluate?

5. Why do $$z=f(x,y)$$ and $$z=g(x,y)=f(x,y)+h$$, for some real number $$h$$, have the same surface area over a region $$R$$?

6. Let $$z=f(x,y)$$ and $$z=g(x,y)=2f(x,y)$$. Why is the surface area of $$g$$ over a region $$R$$ not twice the surface area of $$f$$ over $$R$$?

### Problems

In Exercises 7-10, set up the iterated integral that computes the surfaces area of the given surface over the region $$R$$.

7. $$f(x,y)=\sin x \cos y;\quad R$$ is the rectangle with bounds $$0\le x\le 2\pi$$, $$0\le y \le 2\pi$$.

8. $$f(x,y)=\frac{1}{x^2+y^2+1};\quad R$$ is the circle $$x^2+y^2=9$$.

9. $$f(x,y)=x^2-y^2;\quad R$$ is the rectangle with opposite corners $$(-1,-1)$$ and $$1,1)$$.

10. $$f(x,y)=\frac{1}{e^{x^2}+1};\quad R$$ is the rectangle bounded by $$-5\le x \le 5$$ and $$0\le y \le 1$$.

In Exercises 11-19, find the area of the given surface over the region $$R$$.

11. $$f(x,y)=3x-7y+2;\quad R$$ is the rectangle with opposite corners $$(-1,0)\text{ and }(1,3)$$.

12. $$f(x,y)=2x+2y+2;\quad R$$ is the triangle with corners $$(0,0),(1,0)\text{ and }(0,1)$$.

13. $$f(x,y)=x^2+y^2+10;\quad R$$ is the circle $$x^2+y^2=16$$.

14. $$f(x,y)=-2x+4y^2+7\text{ over } R$$, the triangle bounded by $$y=-x,y=x,0\le y \le 1$$.

15. $$f(x,y)=x^2+y$$ over $$R$$, the triangle bounded by $$y=2x,y=0 \text{ and }x=2$$.

16. $$f(x,y)=\frac{2}{3}x^{3/2}$$ over $$R$$, the rectangle with opposite corners $$(0,0)\text{ and }(1,1)$$.

17. $$f(x,y)=10-2\sqrt{x^2+y^2}$$ over $$R$$, the circle $$x^2+y^2=25$$. (This is the cone with height 10 and base radius 5; be sure to compare your result with the known formula.)

18. Find the surface area of the sphere with radius 5 by doubling the surface area of $$f(x,y)=\sqrt{25-x^2-y^2}$$ over $$R$$, the circle $$x^2+y^2=25$$. (Be sure to compare your result with the known formula.)

19. Find the surface area of the ellipse formed by restricting the plane $$f(x,y)=cx+dy+h$$ to the region $$R$$, the circle $$x^2+y^2=1$$, where $$c$$, $$d$$ and $$h$$ are some constants. Your answer should be given in terms of $$c$$ and $$d$$; why does the value of $$h$$ not matter?

## 13.6: Volume Between Surfaces and Triple Integration

### Terms and Concepts

1. The strategy for establishing bounds for triple integrals is "________ to ________, _________ and __________ to _______."

2. Give an informal interpretation of what $$"\int\int\int_D \,dV$$" means.

3. Give two uses of triple integration.

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

### Problems

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 double integral that finds the volume between these surfaces over $$R$$.

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

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

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

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

In Exercises 9-16, a domain $$D$$ is described by its bounding surfaces, along with a graph. Set up the triple integrals that give the volume of $$D$$ in all 6 orders of integration, and find the volume of $$D$$ by evaluating the indicated triple integral.

9. $$D$$ is bounded by the coordinate planes and $$z=2-2x/3-2y$$.
Evaluate the triple integral with order dz dy dz.

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

11. $$D$$ is bounded by the planes $$x=0,x=2,z=-y\text{ and by }z=y^2/2$$.
Evaluate the triple integral with order dy dz dx.

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

13. $$D$$ is bounded by the planes $$x=2,y=1,z=0\text{ and }z=2x+4y-4$$.
Evaluate the triple integral with order dx dy dz.

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

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

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

In Exercises 17-20, evaluate the triple integral.

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

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

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

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

In Exercises 21-24, find the center of mass of the solid represented by the indicated space region $$D$$ with density function $$\delta (x,y,z)$$.

21. $$D$$ is bounded by the coordinate planes and $$z=2-2x/3-2y$$; $$\delta (x,y,z)=10$$g/cm$$^3$$.
(Note: this is the same region as used in Exercise 9.)

22. $$D$$ is bounded by the planes $$y=0,y=2,x=1,z=0 \text{ and }z=(3-x)/2$$; $$\delta (x,y,z)=2$$g/cm$$^3$$.
(Note: this is the same region as used in Exercise 10.)

23. $$D$$ is bounded by the planes $$x=2,y=1,z=0\text{ and }z=2x+4y-4$$; $$\delta (x,y,z)=x^2$$lb/in$$^3$$.
(Note: this is the same region as used in Exercise 13.)

24. $$D$$ is bounded by the planes $$z=2y\text{ and by }y=4-x^2$$. $$\delta (x,y,z)=y^2$$lb/in$$^3$$.
(Note: this is the same region as used in Exercise 14.)

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