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Calculus

James Stewart

Chapter 15

Multiple Integrals - all with Video Answers

Educators

Section 1

15.1 Exercises

02:20

Problem 1

(a) Estimate the volume of the solid that lies below the surface $z=x y$ and above the rectangle

$$
R=\{(x, y) \mid 0 \leqslant x \leqslant 6,0 \leqslant y \leqslant 4\}
$$

Use a Riemann sum with $m=3, n=2$, 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).

James Kiss
James Kiss
Numerade Educator
11:50

Problem 2

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

Shu Naito
Shu Naito
Numerade Educator
02:47

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 comers.
(b) Use the Midpoint Rule to estimate the integral in part (a).

James Kiss
James Kiss
Numerade Educator
04:15

Problem 4

(a) Estimate the volume of the solid that lies below the surface $z=1+x^2+3 y$ and above the rectangle $R=[1,2] \times[0,3]$. Use a Riemann sum with $m=n=2$ and choose the sample points to be lower left corners.
(b) Use the Midpoint Rule to estimate the volume in part (a).

Linda Hand
Linda Hand
Numerade Educator
05:05

Problem 5

A table of values is given for a function $f(x, y)$ defined on $R=[0,4] \times[2,4]$.
(a) Estimate $\iint_R f(x, y) d A$ using the Midpoint Rule with $m=n=2$.
(b) Estimate the double integral with $m=n=4$ by choosing the sample points to be the points closest to the origin.
TABLE CAN'T COPY.

Linda Hand
Linda Hand
Numerade Educator
04:56

Problem 6

A $20-\mathrm{ft}-\mathrm{by}-30$-ft swimming pool is filled with water. The depth is measured at 5 -ft intervals, starting at one corner of the pool, and the values are recorded in the table. Estimate the volume of water in the pool.
TABLE CAN'T COPY.

Linda Hand
Linda Hand
Numerade Educator
02:36

Problem 7

Let $V$ be the volume of the solid that lies under the graph of $f(x, y)=\sqrt{52-x^2-y^2}$ and above the rectangle given by $2 \leqslant x \leqslant 4,2 \leqslant y \leqslant 6$. We use the lines $x=3$ and $y=4$ to divide $R$ into subrectangles. Let $L$ and $U$ be the Riemann sums computed using lower left corners and upper right corners, respectively. Without calculating the numbers $V, L$, and $U$, arrange them in increasing order and explain your reasoning.

Linda Hand
Linda Hand
Numerade Educator
01:55

Problem 8

The figure shows level curves of a function $f$ in the square $R=[0,2] \times[0,2]$. Use the Midpoint Rule with $m=n=2$ to estimate $\iint_R f(x, y) d A$. How could you improve your estimate?

Linda Hand
Linda Hand
Numerade Educator
07:32

Problem 9

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

Shu Naito
Shu Naito
Numerade Educator
05:55

Problem 10

The contour map shows the temperature, in degrees Fahrenheit, at $4: 00 \mathrm{PM}$ on February 26, 2007, in Colorado. (The state measures 388 mi west to east and 276 mi south to north.) Use the Midpoint Rule with $m=n=4$ to estimate the average temperature in Colorado at that time.

ag
Alan Ghazarians
Numerade Educator
03:23

Problem 11

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

Christian Otero
Christian Otero
Numerade Educator
05:04

Problem 12

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

Christian Otero
Christian Otero
Numerade Educator
04:38

Problem 13

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

Christian Otero
Christian Otero
Numerade Educator
03:17

Problem 14

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.

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
03:13

Problem 15

Use a programmable calculator or computer (or the sum command on a CAS) to estimate

$$
\iint_R \sqrt{1+x e^{-y}} d A
$$

where $R=[0,1] \times[0,1]$. Use the Midpoint Rule with the following numbers of squares of equal size: $1,4,16,64,256$, and 1024 .

Lucas Finney
Lucas Finney
Numerade Educator

Problem 16

Repeat Exercise 15 for the integral $\iint_{\Omega} \sin (x+\sqrt{y}) d A$.

Check back soon!
02:16

Problem 17

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

$$
\iint_R k d A=k(b-a)(d-c)
$$

William Semus
William Semus
Numerade Educator
08:07

Problem 18

Use the result of Exercise 17 to show that

$$
0 \leqslant \iint_\pi \sin \pi x \cos \pi y d A \leqslant \frac{1}{32}
$$

where $R=\left[0, \frac{1}{4}\right] \times\left[\frac{1}{4}, \frac{1}{2}\right]$.

CO
Chizoba Osajindu
Numerade Educator