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Chapter 12

MULTIPLE INTEGRALS

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Problem 1

(a) Estimate the volume of the solid that lies below the surface $z=x y$ and above the rectangle
$$R=\{(x, y) | 0 \leqslant x \leqslant 6,0 \leqslant y \leqslant 4\}$$
Use a Riemann sum with $m=3, n=2,$ and a regular partition, and take the sample point to be the upper right corner of each square.
(b) Use the Midpoint Rule to estimate the volume of the solid in part (a).

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Problem 2

If $R=[0,4] \times[-1,2],$ use a Riemann sum with $m=2,$ $n=3$ to estimate the value of $\iint_{R}\left(1-x y^{2}\right)$ $dA$. Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.

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Problem 3

(a) Use a Riemann sum with $m=n=2$ to estimate the value of $\iint_{R} x e^{-x y} d A,$ where $R=[0,2] \times[0,1] .$ Take the sample points to be upper right corners.
(b) Use the Midpoint Rule to estimate the integral in part (a).

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Problem 4

(a) Estimate the volume of the solid that lies below the surface $z=x+2 y^{2}$ and above the rectangle $R=[0,2] \times[0,4] .$ Use a Riemann sum with $m=n=2$ and choose the sample points to be lower
right corners.
(b) Use the Midpoint Rule to estimate the volume in part (a).
(c) Evaluate the double integral and compare your answer with the estimates in parts (a) and (b).

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Problem 5

A contour map is shown for a function $f$ on the square $R=[0,4] \times[0,4] .$ Use the Midpoint Rule with $m=n=2$ to estimate the value of $\iint_{R} f(x, y) d A.$

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Problem 6

$\mathrm{A} 20-\mathrm{ft}-\mathrm{by}-30-\mathrm{ft}$swimming pool is filled with water. The depth is measured at 5 -foot intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline & {0} & {5} & {10} & {15} & {20} & {25} & {30} \\ \hline 0 & {2} & {3} & {4} & {6} & {7} & {8} & {8} \\ {5} & {2} & {3} & {4} & {7} & {8} & {10} & {8} \\ {10} & {2} & {4} & {6} & {8} & {10} & {12} & {10} \\ {15} & {2} & {3} & {4} & {5} & {6} & {8} & {7} \\ {20} & {2} & {2} & {2} & {2} & {3} & {4} & {4} \\ \hline\end{array}$$

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Problem 7

Evaluate the double integral by first identifying it as the volume of a solid.
$\iint_{R} 3 d A, \quad R=\{(x, y) |-2 \leqslant x \leqslant 2,1 \leqslant y \leqslant 6\}$

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Problem 8

Evaluate the double integral by first identifying it as the volume of a solid.
$\iint_{R}(5-x) d A, \quad R=\{(x, y) | 0 \leqslant x \leqslant 5,0 \leqslant y \leqslant 3\}$

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Problem 9

Evaluate the double integral by first identifying it as the volume of a solid.
$\iint_{R}(4-2 y) d A, \quad R=[0,1] \times[0,1]$

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Problem 10

The integral $\iint_{R} \sqrt{9-y^{2}} d A,$ where $R=[0,4] \times[0,2]$ represents the volume of a solid. Sketch the solid.

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Problem 11

Calculate the iterated integral.
$\int_{1}^{4} \int_{0}^{2}\left(6 x^{2} y-2 x\right) d y d x$

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Problem 12

Calculate the iterated integral.
$\int_{0}^{1} \int_{1}^{2}\left(4 x^{3}-9 x^{2} y^{2}\right) d y d x$

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Problem 13

Calculate the iterated integral.
$\int_{0}^{2} \int_{0}^{4} y^{3} e^{2 x} d y d x$

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Problem 14

Calculate the iterated integral.
$\int_{1}^{3} \int_{1}^{5} \frac{\ln y}{x y} d y d x$

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Problem 15

Calculate the iterated integral.
$\int_{-3}^{3} \int_{0}^{\pi / 2}\left(y+y^{2} \cos x\right) d x d y$

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Problem 16

Calculate the iterated integral.
$\int_{0}^{1} \int_{0}^{3} e^{x+3 y} d x d y$

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Problem 17

Calculate the iterated integral.
$\int_{1}^{4} \int_{1}^{2}\left(\frac{x}{y}+\frac{y}{x}\right) d y d x$

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Problem 18

Calculate the iterated integral.
$\int_{0}^{1} \int_{0}^{1} \sqrt{s+t} d s d t$

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Problem 19

Calculate the iterated integral.
$\int_{0}^{1} \int_{0}^{1} v\left(u+v^{2}\right)^{4} d u d v$

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Problem 20

Calculate the iterated integral.
$\int_{0}^{1} \int_{0}^{1} x y \sqrt{x^{2}+y^{2}} d y d x$

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Problem 21

Calculate the double integral.
$\iint_{R} \frac{x y^{2}}{x^{2}+1} d A, \quad R=\{(x, y) | 0 \leqslant x \leqslant 1,-3 \leqslant y \leqslant 3\}$

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Problem 22

Calculate the double integral.
$\iint_{R}\left(y+x y^{-2}\right) d A, \quad R=\{(x, y) | 0 \leqslant x \leqslant 2,1 \leqslant y \leqslant 2\}$

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Problem 23

Calculate the double integral.
$\iint_{R} x \sin (x+y) d A, \quad R=[0, \pi / 6] \times[0, \pi / 3]$

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Problem 24

Calculate the double integral.
$\iint_{R} \frac{1+x^{2}}{1+y^{2}} d A, \quad R=\{(x, y) | 0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1\}$

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Problem 25

Calculate the double integral.
$\iint_{R} y e^{-x y} d A, \quad R=[0,2] \times[0,3]$

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Problem 26

Calculate the double integral.
$\iint_{R} \frac{x}{1+x y} d A, \quad R=[0,1] \times[0,1]$

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Problem 27

Sketch the solid whose volume is given by the iterated integral.
$\int_{0}^{1} \int_{0}^{1}(4-x-2 y) d x d y$

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Problem 28

Sketch the solid whose volume is given by the iterated integral.
$\int_{0}^{1} \int_{0}^{1}\left(2-x^{2}-y^{2}\right) d y d x$

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Problem 29

Find the volume of the solid that lies under the plane $4 x+6 y-2 z+15=0$ and above the rectangle $R=\{(x, y) |-1 \leqslant x \leqslant 2,-1 \leqslant y \leqslant 1\}.$

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Problem 30

Find the volume of the solid that lies under the hyperbolic paraboloid $z=3 y^{2}-x^{2}+2$ and above the rectangle $R=[-1,1] \times[1,2].$

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Problem 31

Find the volume of the solid lying under the elliptic paraboloid $x^{2} / 4+y^{2} / 9+z=1$ and above the rectangle $R=[-1,1] \times[-2,2].$

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Problem 32

Find the volume of the solid enclosed by the surface $z=1+e^{x} \sin y$ and the planes $x=\pm 1, y=0, y=\pi$ and $z=0.$

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Problem 33

Find the volume of the solid enclosed by the surface $z=x \sec ^{2} y$ and the planes $z=0, x=0, x=2, y=0$ and $y=\pi / 4.$

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Problem 34

Find the volume of the solid in the first octant bounded by the cylinder $z=16-x^{2}$ and the plane $y=5 .$

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Problem 35

Find the volume of the solid enclosed by the paraboloid $z=2+x^{2}+(y-2)^{2}$ and the planes $z=1, x=1$ $x=-1, y=0,$ and $y=4.$

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Problem 36

Graph the solid that lies between the surface $z=2 x y /\left(x^{2}+1\right)$ and the plane $z=x+2 y$ and is bounded by the planes $x=0, x=2, y=0,$ and $y=4$ Then find its volume.

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Problem 37

Use a computer algebra system to find the exact value of the integral $\iint_{R} x^{5} y^{3} e^{x y} d A,$ where $R=[0,1] \times[0,1] .$ Then use the CAS to draw the solid whose volume is given by the integral.

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Problem 38

Graph the solid that lies between the surfaces $z=e^{-x^{2}} \cos \left(x^{2}+y^{2}\right)$ and $z=2-x^{2}-y^{2}$ for $|x| \leqslant 1$ $|y| \leqslant 1 .$ Use a computer algebra system to approximate the volume of this solid correct to four decimal places.

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Problem 39

The average value of a function $f(x, y)$ over a rectangle $R$ is defined to be
$$f_{\mathrm{ave}}=\frac{1}{A(R)} \iint_{R} f(x, y) d A$$
(Compare with the definition for functions of one variable in Section $5.4 .$) . Find the average value of $f$ over the given rectangle.
$f(x, y)=x^{2} y$ $R$ has vertices $(-1,0),(-1,5),(1,5),(1,0)$

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Problem 40

The average value of a function $f(x, y)$ over a rectangle $R$ is defined to be
$$f_{\mathrm{ave}}=\frac{1}{A(R)} \iint_{R} f(x, y) d A$$
(Compare with the definition for functions of one variable in Section $5.4 .$) . Find the average value of $f$ over the given rectangle.
$f(x, y)=e^{y} \sqrt{x+e^{y}}, \quad R=[0,4] \times[0,1]$

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Problem 41

If $f$ is a constant function, $f(x, y)=k,$ and $R=[a, b] \times[c, d],$ show that
$$\iint_{R} k d A=k(b-a)(d-c)$$

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Problem 42

Use the result of Exercise 41 to show that
$$0 \leqslant \iint_{R} \sin \pi x \cos \pi y d A \leqslant \frac{1}{32}$$
where $R=\left[0, \frac{1}{4}\right] \times\left[\frac{1}{4}, \frac{1}{2}\right]$

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Problem 43

Use symmetry to evaluate the double integral.
$\iint_{R} \frac{x y}{1+x^{4}} d A, \quad R=\{(x, y) |-1 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1\}$

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Problem 44

Use symmetry to evaluate the double integral.
$\iint_{R}\left(1+x^{2} \sin y+y^{2} \sin x\right) d A, \quad R=[-\pi, \pi] \times[-\pi, \pi]$

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Problem 45

Use your CAS to compute the iterated integrals
$$\int_{0}^{1} \int_{0}^{1} \frac{x-y}{(x+y)^{3}} d y d x \quad \text { and } \quad \int_{0}^{1} \int_{0}^{1} \frac{x-y}{(x+y)^{3}} d x d y$$
Do the answers contradict Fubini's Theorem? Explain what is happening.

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Problem 46

(a) In what way are the theorems of Fubini and Clairaut similar?
(b) If $f(x, y)$ is continuous on $[a, b] \times[c, d]$ and
$$g(x, y)=\int_{a}^{x} \int_{c}^{y} f(s, t) d t d s$$
for $a< x < b, c < y < d,$ show that $g_{x y}=g_{y x}=f(x, y).$

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