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Thomas Calculus

George B. Thomas, Jr.

Chapter 15

Multiple Integrals

Educators


Problem 1

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{1}^{2} \int_{0}^{4} 2 x y d y d x
$$

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Problem 2

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{0}^{2} \int_{-1}^{1}(x-y) d y d x
$$

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Problem 3

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y
$$

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Problem 4

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{0}^{1} \int_{0}^{1}\left(1-\frac{x^{2}+y^{2}}{2}\right) d x d y
$$

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Problem 5

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{0}^{3} \int_{0}^{2}\left(4-y^{2}\right) d y d x
$$

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Problem 6

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{0}^{3} \int_{-2}^{0}\left(x^{2} y-2 x y\right) d y d x
$$

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Problem 7

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{0}^{1} \int_{0}^{1} \frac{y}{1+x y} d x d y
$$

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Problem 8

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{1}^{4} \int_{0}^{4}\left(\frac{x}{2}+\sqrt{y}\right) d x d y
$$

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Problem 9

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{0}^{\ln 2} \int_{1}^{\ln 5} e^{2 x+y} d y d x
$$

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Problem 10

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{0}^{1} \int_{1}^{2} x y e^{x} d y d x
$$

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Problem 11

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{-1}^{2} \int_{0}^{\pi / 2} y \sin x d x d y
$$

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Problem 12

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{\pi}^{2 \pi} \int_{0}^{\pi}(\sin x+\cos y) d x d y
$$

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Problem 13

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{1}^{4} \int_{1}^{e} \frac{\ln x}{x y} d x d y
$$

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Problem 14

In Exercises $1-14,$ evaluate the iterated integral.
$$
\int_{-1}^{2} \int_{1}^{2} x \ln y d y d x
$$

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Problem 15

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R}\left(6 y^{2}-2 x\right) d A, \quad R : 0 \leq x \leq 1, \quad 0 \leq y \leq 2
$$

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Problem 16

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R}\left(\frac{\sqrt{x}}{y^{2}}\right) d A, \quad R : \quad 0 \leq x \leq 4, \quad 1 \leq y \leq 2
$$

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Problem 17

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R} x y \cos y d A, \quad R :-1 \leq x \leq 1, \quad 0 \leq y \leq \pi
$$

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Problem 18

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R} y \sin (x+y) d A, \quad R :-\pi \leq x \leq 0, \quad 0 \leq y \leq \pi
$$

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Problem 19

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R} e^{x-y} d A, \quad R : \quad 0 \leq x \leq \ln 2, \quad 0 \leq y \leq \ln 2
$$

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Problem 20

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R} x y e^{\mathrm{y}^{2}} d A, \quad R : \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 1
$$

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Problem 21

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R} \frac{x y^{3}}{x^{2}+1} d A, \quad R : \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2
$$

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Problem 22

In Exercises $15-22,$ evaluate the double integral over the given region R
$$
\iint_{R} \frac{y}{x^{2} y^{2}+1} d A, \quad R : \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1
$$

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Problem 23

In Exercises 23 and $24,$ integrate $f$ over the given region.
$$
\begin{array}{l}{\text { Square } f(x, y)=1 /(x y) \quad \text { over } \quad \text { the square } \quad 1 \leq x \leq 2} \\ {1 \leq y \leq 2}\end{array}
$$

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Problem 24

In Exercises 23 and $24,$ integrate $f$ over the given region.
$$
\begin{array}{l}{\text { Rectangle } f(x, y)=y \cos x y \text { over the rectangle } 0 \leq x \leq \pi} \\ {0 \leq y \leq 1}\end{array}
$$

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Problem 25

Find the volume of the region bounded above by the paraboloid
$z=x^{2}+y^{2}$ and below by the square $R :-1 \leq x \leq 1$ ,
$-1 \leq y \leq 1$

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Problem 26

Find the volume of the region bounded above by the elliptical
paraboloid $z=16-x^{2}-y^{2}$ and below by the square
$R : 0 \leq x \leq 2,0 \leq y \leq 2$

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Problem 27

Find the volume of the region bounded above by the plane
$z=2-x-y$ and below by the square $R : 0 \leq x \leq 1$
$0 \leq y \leq 1 .$

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Problem 28

Find the volume of the region bounded above by the plane
$z=y / 2$ and below by the rectangle $R : 0 \leq x \leq 4,0 \leq y \leq 2$

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Problem 29

Find the volume of the region bounded above by the surface
$z=2 \sin x \cos y$ and below by the rectangle $R : 0 \leq x \leq \pi / 2$
$0 \leq y \leq \pi / 4$

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Problem 30

Find the volume of the region bounded above by the surface
$z=4-y^{2} \quad$ and $\quad$ below by the rectangle $\quad R : 0 \leq x \leq 1$
$0 \leq y \leq 2 .$

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Problem 31

Find a value of the constant $k$ so that $\int_{1}^{2} \int_{0}^{3} k x^{2} y d x d y=1$

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Problem 32

Evaluate $$\int_{-1}^{1} \int_{0}^{\pi / 2} x \sin \sqrt{y} d y d x$$

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Problem 33

Use Fubini's Theorem to evaluate
$$\int_{0}^{2} \int_{0}^{1} \frac{x}{1+x y} d x d y$$

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Problem 34

Use Fubini's Theorem to evaluate
$$\int_{0}^{1} \int_{0}^{3} x e^{x y} d x d y$$

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Problem 35

Use a software application to compute the integrals
$$ { a, }\int_{0}^{1} \int_{0}^{2} \frac{y-x}{(x+y)^{3}} d x d y$$
$$ { b. }\int_{0}^{2} \int_{0}^{1} \frac{y-x}{(x+y)^{3}} d y d x$$
Explain why your results do not contradict Fubini's Theorem.

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Problem 36

If $f(x, y)$ is continuous over $R : a \leq x \leq b, c \leq y \leq d$ and
$$F(x, y)=\int_{a}^{x} \int_{c}^{y} f(u, v) d v d u$$
on the interior of $R,$ find the second partial derivatives $F_{x y}$ and $F_{y x}$

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