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Chapter 15

MULTIPLE INTEGRALS

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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 comer of each square.
(b) Use the Midpoint Rule to estimate the volume of the solid
in part (a).

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Problem 2

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

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Problem 3

(a) Use a Riemann sum with $m=n=2$ to estimate the value
of $\iint_{R} \sin (x+y) d A,$ where $R=[0, \pi] \times[0, \pi]$ . Take the
sample points to be lower left comers.
(b) Use the Midpoint Rule to estimate the integral in part (a).

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Problem 4

(a) Estimate the volume of the solid that lies below the surface
$z=x+2 y^{2}$ and above the rectangle $R=[0,2] \times[0,4]$
Use a Riemann sum with $m=n=2$ and choose the
sample points to be lower right corners.
(b) Use the Midpoint Rule to estimate the volume in part (a).

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Problem 5

A table of values is given for a function $f(x, y)$ defined on
$R=[1,3] \times[0,4] .$
(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 farthest from the origin.

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Problem 6

A $20-\mathrm{ft}$ -by- 30 -ft swimming pool is filled with water. The depth
is measured at $5-\mathrm{ft}$ intervals, starting at one corner of the pool,
and the values are recorded in the table. Estimate the volume of
water in the pool.

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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
$2 \leq x \leqslant 4,2 \leqslant y \leqslant y \leqslant 6 .$ 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.

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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?

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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 $f$

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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 east to west and 276 mi north to south.) Use
the Midpoint Rule with $m=n=4$ to estimate the average
temperature in Colorado at that time.

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Problem 11

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

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Problem 12

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

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Problem 13

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

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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.

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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 $1024 .$

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Problem 16

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

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Problem 17

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

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

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