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Calculus Early Transcendentals

James Stewart

Chapter 15

Multiple Integrals - all with Video Answers

Educators

Section 1

Double Integrals over Rectangles

04:05

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 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).

Linda Hand
Linda Hand
Numerade Educator
04:43

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 comers and (b) the upper left corners of the rectangles.

Linda Hand
Linda Hand
Numerade Educator
03:21

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).

Linda Hand
Linda Hand
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 $\quad m=n=2$
(b) Estimate the double integral with $m=n=4$ by choosing the sample points to be the points closest to the origin.

Linda Hand
Linda Hand
Numerade Educator
04:37

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.
$$\begin{array}{|c|c|c|c|c|c|c|}\hline & {0} & {5} & {10} & {15} & {20} & {25} & {30} \\ \hline 0 & {2} & {3} & {4} & {6} & {7} & {8} & {8} \\ \hline 5 & {2} & {3} & {4} & {7} & {8} & {10} & {8} \\ \hline 10 & {2} & {4} & {6} & {8} & {10} & {12} & {10} \\ \hline 15 & {2} & {3} & {4} & {5} & {6} & {8} & {7} \\ \hline 20 & {2} & {2} & {2} & {2} & {3} & {4} & {4} \\ \hline\end{array}$$

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
02:16

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 $\int_{R} f(x, y) d A$ .
(b) Estimate the average value of $f$.

Linda Hand
Linda Hand
Numerade Educator
05:55

Problem 10

The contour map shows the temperature, in degrees Fahrenheit, at $4 : 00$ 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.
$$\iint_{R} 3 d A, \quad R=\{(x, y) |-2 \leqslant x \leqslant 2,1 \leqslant y \leqslant 6\}$$

Christian Otero
Christian Otero
Numerade Educator
05:04

Problem 12

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

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
03:01

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.

Christian Otero
Christian Otero
Numerade Educator
03:21

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 $1024 .$

ag
Alan Ghazarians
Numerade Educator
03:20

Problem 16

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

Nolwazi Dube
Nolwazi Dube
Numerade Educator
03:01

Problem 17

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

Christian Otero
Christian Otero
Numerade Educator
08:07

Problem 18

Use the result of Exercise 17 to show that
$$0 \leqslant \iint_{R} \sin \pi x \cos \pi y d A \leqslant \frac{1}{32}$$
where $R=\left[0, \frac{1}{4}\right] \times\left[\frac{1}{4}, \frac{1}{2}\right]$.

CO
Chizoba Osajindu
Numerade Educator