3.E: Chapter 3 Review Exercises

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.)