University Calculus: Early Transcendentals 4th

Joel Hass, Christopher Heil, Przemyslaw Bogacki

Chapter 14

Multiple Integrals

Educators


Problem 1

Evaluate the iterated integral.
$$\int_{1}^{2} \int_{0}^{4} 2 x y d y d x$$

Bobby B.
University of North Texas

Problem 2

Evaluate the iterated integral.
$$\int_{0}^{2} \int_{-1}^{1}(x-y) d y d x$$

Bobby B.
University of North Texas

Problem 3

Evaluate the iterated integral.
$$\int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y$$

Bobby B.
University of North Texas

Problem 4

Evaluate the iterated integral.
$$\int_{0}^{1} \int_{0}^{1}\left(1-\frac{x^{2}+y^{2}}{2}\right) d x d y$$

Bobby B.
University of North Texas

Problem 5

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

Bobby B.
University of North Texas

Problem 6

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

Bobby B.
University of North Texas

Problem 7

Evaluate the iterated integral.
$$\int_{0}^{1} \int_{0}^{1} \frac{y}{1+x y} d x d y$$

Bobby B.
University of North Texas

Problem 8

Evaluate the iterated integral.
$$\int_{1}^{4} \int_{0}^{4}\left(\frac{x}{2}+\sqrt{y}\right) d x d y$$

Bobby B.
University of North Texas

Problem 9

Evaluate the iterated integral.
$$\int_{0}^{\ln 2} \int_{1}^{\ln 5} e^{2 x+y} d y d x$$

Bobby B.
University of North Texas

Problem 10

Evaluate the iterated integral.
$$\int_{0}^{1} \int_{1}^{2} x y e^{x} d y d x$$

Bobby B.
University of North Texas

Problem 11

Evaluate the iterated integral.
$$\int_{-1}^{2} \int_{0}^{\pi / 2} y \sin x d x d y$$

Bobby B.
University of North Texas

Problem 12

Evaluate the iterated integral.
$$\int_{\pi}^{2 \pi} \int_{0}^{\pi}(\sin x+\cos y) d x d y$$

Bobby B.
University of North Texas

Problem 13

Evaluate the iterated integral.
$$\int_{1}^{4} \int_{1}^{e} \frac{\ln x}{x y} d x d y$$

Bobby B.
University of North Texas

Problem 14

Evaluate the iterated integral.
$$\int_{-1}^{2} \int_{1}^{2} x \ln y d y d x$$

Bobby B.
University of North Texas

Problem 15

Find all values of the constant $c$ so that $\int_{0}^{1} \int_{0}^{c}(2 x+y) d x d y=3$.

Bobby B.
University of North Texas

Problem 16

Find all values of the constant $c$ so that $\int_{-1}^{c} \int_{0}^{2}(x y+1) d y d x=4+4 c$.

Bobby B.
University of North Texas

Problem 17

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

Bobby B.
University of North Texas

Problem 18

Evaluate the double integral over the given region $R$.
$$\iint_{R}\left(\frac{\sqrt{x}}{y^{2}}\right) d A, \quad R: \quad 0 \leq x \leq 4, \quad 1 \leq y \leq 2$$

Bobby B.
University of North Texas

Problem 19

Evaluate the double integral over the given region $R$.
$$\iint_{R} x y \cos y d A, \quad R: \quad-1 \leq x \leq 1, \quad 0 \leq y \leq \pi$$

Bobby B.
University of North Texas

Problem 20

Evaluate the double integral over the given region $R$.
$$\iint_{R} y \sin (x+y) d A, \quad R: \quad-\pi \leq x \leq 0, \quad 0 \leq y \leq \pi$$

Bobby B.
University of North Texas

Problem 21

Evaluate the double integral over the given region $R$.
$$\iint_{R} e^{x-y} d A, \quad R: \quad 0 \leq x \leq \ln 2, \quad 0 \leq y \leq \ln 2$$

Bobby B.
University of North Texas

Problem 22

Evaluate the double integral over the given region $R$.
$$\iint_{R} x y e^{v^{2}} d A, \quad R: \quad 0 \leq x \leq 2, \quad 0 \leq y \leq 1$$

Bobby B.
University of North Texas

Problem 23

Evaluate the double integral over the given region $R$.
$$\iint_{R} \frac{x y^{3}}{x^{2}+1} d A, \quad R: \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 2$$

Bobby B.
University of North Texas

Problem 24

Evaluate the double integral over the given region $R$.
$$\iint_{R} \frac{y}{x^{2} y^{2}+1} d A, \quad R: \quad 0 \leq x \leq 1, \quad 0 \leq y \leq 1$$

Bobby B.
University of North Texas

Problem 25

Integrate $f$ over the given region.
Square $\quad f(x, y)=1 /(x y)$ over the square $1 \leq x \leq 2$ $1 \leq y \leq 2$

Bobby B.
University of North Texas

Problem 26

Integrate $f$ over the given region.
Rectangle $f(x, y)=y \cos x y$ over the rectangle $0 \leq x \leq \pi$ $0 \leq y \leq 1$

Bobby B.
University of North Texas

Problem 27

Sketch the solid whose volume is given by the specified integral.
$\int_{0}^{1} \int_{0}^{2}\left(9-x^{2}-y^{2}\right) d y d x$

Bobby B.
University of North Texas

Problem 28

Sketch the solid whose volume is given by the specified integral.
$\int_{0}^{3} \int_{1}^{4}(7-x-y) d x d y$

Bobby B.
University of North Texas

Problem 29

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

Bobby B.
University of North Texas

Problem 30

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

Bobby B.
University of North Texas

Problem 31

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

Bobby B.
University of North Texas

Problem 32

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

Bobby B.
University of North Texas

Problem 33

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

Bobby B.
University of North Texas

Problem 34

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

Bobby B.
University of North Texas

Problem 35

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

Bobby B.
University of North Texas

Problem 36

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

Bobby B.
University of North Texas

Problem 37

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

Bobby B.
University of North Texas

Problem 38

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

Bobby B.
University of North Texas

Problem 39

Use a software application to compute the integrals
a. $\int_{0}^{1} \int_{0}^{2} \frac{y-x}{(x+y)^{3}} d x d y \quad$ 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.

Bobby B.
University of North Texas

Problem 40

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 $F_{y x}$.

Bobby B.
University of North Texas