Thomas Calculus

George B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano

Chapter 15

Multiple Integrals - all with Video Answers

Educators

Section 1

Double Integrals

Problem 1

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{0}^{3} \int_{0}^{2}\left(4-y^{2}\right) d y d x
$$

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Problem 2

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{0}^{3} \int_{-2}^{0}\left(x^{2} y-2 x y\right) d y d x
$$

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Problem 3

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y
$$

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Problem 4

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{\pi}^{2 \pi} \int_{0}^{\pi}(\sin x+\cos y) d x d y
$$

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Problem 5

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{0}^{\pi} \int_{0}^{x} x \sin y d y d x
$$

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Problem 6

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{0}^{\pi} \int_{0}^{\sin x} y d y d x
$$

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Problem 7

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{1}^{\ln 8} \int_{0}^{\ln y} e^{x+y} d x d y
$$

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Problem 8

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{1}^{2} \int_{y}^{y^{2}} d x d y
$$

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Problem 9

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{0}^{1} \int_{0}^{y^{2}} 3 y^{3} e^{x y} d x d y
$$

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Problem 10

In Exercises $1-10,$ sketch the region of integration and evaluate the integral.
$$
\int_{1}^{4} \int_{0}^{\sqrt{x}} \frac{3}{2} e^{y / \sqrt{x}} d y d x
$$

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Problem 11

In Exercises $11-16,$ integrate $f$ over the given region.
Quadrilateral $f(x, y)=x / y$ over the region in the first quadrant bounded by the lines $y=x, y=2 x, x=1, x=2$

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Problem 12

In Exercises $11-16,$ integrate $f$ over the given region.
$\begin{array}{ll}{\text { Square } f(x, y)=1 /(x y)} & {\text { over } \quad \text { the square } \quad 1 \leq x \leq 2} \\ {1 \leq y \leq 2}\end{array}$

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Problem 13

In Exercises $11-16,$ integrate $f$ over the given region.
Triangle $f(x, y)=x^{2}+y^{2}$ over the triangular region with vertices $(0,0),(1,0),$ and $(0,1)$

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Problem 14

In Exercises $11-16,$ integrate $f$ over the given region.
Rectangle $f(x, y)=y \cos x y$ over the rectangle $0 \leq x \leq \pi$ $0 \leq y \leq 1$

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Problem 15

In Exercises $11-16,$ integrate $f$ over the given region.
Triangle $f(u, v)=v-\sqrt{u}$ over the triangular region cut from the first quadrant of the $u v$ -plane by the line $u+v=1$

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Problem 16

In Exercises $11-16,$ integrate $f$ over the given region.
Curved region $f(s, t)=e^{s} \ln t$ over the region in the first quadrant of the $s t$ -plane that lies above the curve $s=\ln t$ from $t=1$ to $t=2$

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Problem 17

Each of Exercises $17-20$ gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral.
$$
\int_{-2}^{0} \int_{v}^{v} 2 d p d v \quad(\text { the } p v \text { -plane })
$$

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Problem 18

Each of Exercises $17-20$ gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral.
$$
\int_{0}^{1} \int_{0}^{\sqrt{1-s^{2}}} 8 t d t d s \quad \text { (the } s t-\text { plane } )
$$

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Problem 19

Each of Exercises $17-20$ gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral.
$$
\int_{-\pi / 3}^{\pi / 3} \int_{0}^{\sec t} 3 \cos t d u d t \quad \text { (the } t u-\text { plane } )
$$

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Problem 20

Each of Exercises $17-20$ gives an integral over a region in a Cartesian coordinate plane. Sketch the region and evaluate the integral.
$$
\int_{0}^{3} \int_{1}^{4-2 u} \frac{4-2 u}{v^{2}} d v d u \quad \text { (the } u v-\text { plane } )
$$

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Problem 21

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{1} \int_{2}^{4-2 x} d y d x
$$

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Problem 22

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{2} \int_{y-2}^{0} d x d y
$$

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Problem 23

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{1} \int_{y}^{\sqrt{y}} d x d y
$$

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Problem 24

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{1} \int_{1-x}^{1-x^{2}} d y d x
$$

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Problem 25

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{1} \int_{1}^{e^{x}} d y d x
$$

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Problem 26

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{\ln 2} \int_{e^{x}}^{2} d x d y
$$

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Problem 27

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{3 / 2} \int_{0}^{9-4 x^{2}} 16 x d y d x
$$

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Problem 28

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{2} \int_{0}^{4-y^{2}} y d x d y
$$

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Problem 29

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}} 3 y d x d y
$$

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Problem 30

In Exercises $21-30,$ sketch the region of integration and write an equivalent double integral with the order of integration reversed.
$$
\int_{0}^{2} \int_{-\sqrt{4-x^{2}}}^{\sqrt{4-x^{2}}} 6 x d y d x
$$

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Problem 31

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{\pi} \int_{x}^{\pi} \frac{\sin y}{y} d y d x
$$

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Problem 32

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{2} \int_{x}^{2} 2 y^{2} \sin x y d y d x
$$

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Problem 33

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{1} \int_{y}^{1} x^{2} e^{x y} d x d y
$$

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Problem 34

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{2} \int_{0}^{4-x^{2}} \frac{x e^{2 y}}{4-y} d y d x
$$

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Problem 35

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{2 \sqrt{\ln 3}} \int_{y / 2}^{\sqrt{\ln 3}} e^{x^{2}} d x d y
$$

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Problem 36

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{3} \int_{\sqrt{x / 3}}^{1} e^{y^{3}} d y d x
$$

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Problem 37

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{1 / 16} \int_{y^{1 / 4}}^{1 / 2} \cos \left(16 \pi x^{5}\right) d x d y
$$

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Problem 38

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
$$
\int_{0}^{8} \int_{\sqrt[3]{x}}^{2} \frac{d y d x}{y^{4}+1}
$$

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Problem 39

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
Square region $\iint_{R}\left(y-2 x^{2}\right) d A$ where $R$ is the region bounded by the square $|x|+|y|=1$

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Problem 40

In Exercises $31-40,$ sketch the region of integration, reverse the order of integration, and evaluate the integral.
Triangular region $\prod \int_{R} x y d A$ where $R$ is the region bounded by the lines $y=x, y=2 x,$ and $x+y=2$

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Problem 41

Find the volume of the region bounded by the paraboloid $z=x^{2}+y^{2}$ and below by the triangle enclosed by the lines $y=x, x=0,$ and $x+y=2$ in the $x y$ -plane.

Vipender Yadav

Numerade Educator

Problem 42

Find the volume of the solid that is bounded above by the cylinder $z=x^{2}$ and below by the region enclosed by the parabola $y=2-x^{2}$ and the line $y=x$ in the $x y$ -plane.

Vipender Yadav

Numerade Educator

Problem 43

Find the volume of the solid whose base is the region in the $x y-$ plane that is bounded by the parabola $y=4-x^{2}$ and the line $y=3 x,$ while the top of the solid is bounded by the plane $z=x+4 .$

Vipender Yadav

Numerade Educator

Problem 44

Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder $x^{2}+y^{2}=4,$ and the plane $z+y=3 .$

Vipender Yadav

Numerade Educator

Problem 45

Find the volume of the solid in the first octant bounded by the coordinate planes, the plane $x=3,$ and the parabolic cylinder $z=4-y^{2} .$

Yiming Zhang

Numerade Educator

Problem 46

Find the volume of the solid cut from the first octant by the surface $z=4-x^{2}-y$

Linda Hand

Numerade Educator

Problem 47

Find the volume of the wedge cut from the first octant by the cylinder $z=12-3 y^{2}$ and the plane $x+y=2$

Vipender Yadav

Numerade Educator

Problem 48

Find the volume of the solid cut from the square column $|x|+|y| \leq 1$ by the planes $z=0$ and $3 x+z=3$

Nick Johnson

Numerade Educator

Problem 49

Find the volume of the solid that is bounded on the front and back by the planes $x=2$ and $x=1,$ on the sides by the cylinders $y=\pm 1 / x,$ and above and below by the planes $z=x+1$ and $z=0$

Yiming Zhang

Numerade Educator

Problem 50

Find the volume of the solid bounded on the front and back by the planes $x=\pm \pi / 3,$ on the sides by the cylinders $y=\pm \sec x,$ above by the cylinder $z=1+y^{2},$ and below by the $x y$ -plane.

Linh Vu

Numerade Educator

Problem 51

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals in Exercises $51-54$ as iterated integrals.
$$
\int_{1}^{\infty} \int_{e^{-x}}^{1} \frac{1}{x^{3} y} d y d x
$$

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Problem 52

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals in Exercises $51-54$ as iterated integrals.
$$
\int_{-1}^{1} \int_{-1 / \sqrt{1-x^{2}}}^{1 / \sqrt{1-x^{2}}}(2 y+1) d y d x
$$

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Problem 53

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals in Exercises $51-54$ as iterated integrals.
$$
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)\left(y^{2}+1\right)} d x d y
$$

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Problem 54

Improper double integrals can often be computed similarly to improper integrals of one variable. The first iteration of the following improper integrals is conducted just as if they were proper integrals. One then evaluates an improper integral of a single variable by taking appropriate limits, as in Section $8.8 .$ Evaluate the improper integrals in Exercises $51-54$ as iterated integrals.
$$
\int_{0}^{\infty} \int_{0}^{\infty} x e^{-(x+2 y)} d x d y
$$

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Problem 55

$f(x, y)=x+y$ over the region $R$ bounded above by the semicircle $y=\sqrt{1-x^{2}}$ and below by the $x$ -axis, using the partition $x=-1,-1 / 2,0,1 / 4,1 / 2,1$ and $y=0,1 / 2,1$ with $\left(x_{k}, y_{k}\right)$ the
lower left corner in the $k$ th subrectangle (provided the subrectangle lies within $R )$

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Problem 56

$f(x, y)=x+2 y$ over the region $R$ inside the circle $(x-2)^{2}+(y-3)^{2}=1$ using the partition $x=1,3 / 2,2,5 / 2$ 3 and $y=2,5 / 2,3,7 / 2,4$ with $\left(x_{k}, y_{k}\right)$ the center (centroid) in the $k$ th subrectangle (provided the subrectangle lies within $R )$

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Problem 57

Circular sector Integrate $f(x, y)=\sqrt{4-x^{2}}$ over the smaller sector cut from the disk $x^{2}+y^{2} \leq 4$ by the rays $\theta=\pi / 6$ and $\theta=\pi / 2 .$

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Problem 58

Unbounded region Integrate $f(x, y)=1 /\left[\left(x^{2}-x\right)(y-1)^{2 / 3}\right]$ over the infinite rectangle $2 \leq x<\infty, 0 \leq y \leq 2$

Vipender Yadav

Numerade Educator

Problem 59

Noncircular cylinder A solid right (noncircular) cylinder has its base $R$ in the $x y$ -plane and is bounded above by the paraboloid $z=x^{2}+y^{2} .$ The cylinder's volume is
$$
V=\int_{0}^{1} \int_{0}^{y}\left(x^{2}+y^{2}\right) d x d y+\int_{1}^{2} \int_{0}^{2-y}\left(x^{2}+y^{2}\right) d x d y
$$
Sketch the base region $R$ and express the cylinder's volume as a single iterated integral with the order of integration reversed. Then evaluate the integral to find the volume.

Nick Johnson

Numerade Educator

Problem 60

Converting to a double integral Evaluate the integral
$$
\int_{0}^{2}\left(\tan ^{-1} \pi x-\tan ^{-1} x\right) d x
$$
(Hint: Write the integrand as an integral.)

Clayton Bennett

Clayton Bennett

Numerade Educator

Problem 61

Maximizing a double integral What region $R$ in the $x y$ -plane maximizes the value of
$$
\iint_{R}\left(4-x^{2}-2 y^{2}\right) d A ?
$$
Give reasons for your answer.

Bobby Barnes

University of North Texas

Problem 62

Minimizing a double integral What region $R$ in the $x y$ -plane minimizes the value of
$$
\iint_{R}\left(x^{2}+y^{2}-9\right) d A ?
$$
Give reasons for your answer.

Victor Salazar

Numerade Educator

Problem 63

Is it possible to evaluate the integral of a continuous function $f(x, y)$ over a rectangular region in the $x y$ -plane and get different answers depending on the order of integration? Give reasons for your
answer.

Vipender Yadav

Numerade Educator

Problem 64

How would you evaluate the double integral of a continuous function $f(x, y)$ over the region $R$ in the $x y$ -plane enclosed by the triangle with vertices $(0,1),(2,0),$ and $(1,2) ?$ Give reasons for your answer.

Vipender Yadav

Numerade Educator

Problem 65

Unbounded region Prove that
$$
\begin{aligned} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^{2}-y^{2}} d x d y &=\lim _{b \rightarrow \infty} \int_{-b}^{b} \int_{-b}^{b} e^{-x^{2}-y^{2}} d x d y \\ &=4\left(\int_{0}^{\infty} e^{-x^{2}} d x\right)^{2} \end{aligned}
$$

Victor Salazar

Numerade Educator

Problem 66

Improper double integral Evaluate the improper integral
$$
\int_{0}^{1} \int_{0}^{3} \frac{x^{2}}{(y-1)^{2 / 3}} d y d x
$$

Vipender Yadav

Numerade Educator

Problem 67

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises $67-70 .$
$$
\int_{1}^{3} \int_{1}^{x} \frac{1}{x y} d y d x
$$

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Problem 68

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises $67-70 .$
$$
\int_{0}^{1} \int_{0}^{1} e^{-\left(x^{2}+y^{2}\right)} d y d x
$$

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Problem 69

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises $67-70 .$
$$
\int_{0}^{1} \int_{0}^{1} \tan ^{-1} x y d y d x
$$

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Problem 70

Use a CAS double-integral evaluator to estimate the values of the integrals in Exercises $67-70 .$
$$
\int_{-1}^{1} \int_{0}^{\sqrt{1-x^{2}}} 3 \sqrt{1-x^{2}-y^{2}} d y d x
$$

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Problem 71

Use a CAS double-integral evaluator to find the integrals in Exercises $711-76 .$ Then reverse the order of integration and evaluate, again with a CAS.
$$
\int_{0}^{1} \int_{2 y}^{4} e^{x^{2}} d x d y
$$

Willis James

Numerade Educator

Problem 72

Use a CAS double-integral evaluator to find the integrals in Exercises $711-76 .$ Then reverse the order of integration and evaluate, again with a CAS.
$$
\int_{0}^{3} \int_{x^{2}}^{9} x \cos \left(y^{2}\right) d y d x
$$

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Problem 73

Use a CAS double-integral evaluator to find the integrals in Exercises $711-76 .$ Then reverse the order of integration and evaluate, again with a CAS.
$$
\int_{0}^{2} \int_{y^{3}}^{4 \sqrt{2 y}}\left(x^{2} y-x y^{2}\right) d x d y
$$

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Problem 74

Use a CAS double-integral evaluator to find the integrals in Exercises $711-76 .$ Then reverse the order of integration and evaluate, again with a CAS.
$$
\int_{0}^{2} \int_{0}^{4-y^{2}} e^{x y} d x d y
$$

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Problem 75

Use a CAS double-integral evaluator to find the integrals in Exercises $711-76 .$ Then reverse the order of integration and evaluate, again with a CAS.
$$
\int_{1}^{2} \int_{0}^{x^{2}} \frac{1}{x+y} d y d x
$$

Willis James

Numerade Educator

Problem 76

Use a CAS double-integral evaluator to find the integrals in Exercises $711-76 .$ Then reverse the order of integration and evaluate, again with a CAS.
$$
\int_{1}^{2} \int_{y^{3}}^{8} \frac{1}{\sqrt{x^{2}+y^{2}}} d x d y
$$

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