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Chapter 15

Multiple Integrals

Educators

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Problem 1

(a) Estimate the volume of the solid that lies below the surface $ z = xy $ and above the rectangle
$$ R = \{(x, y) \mid 0 \le x \le 6, 0 \le y \le 4 \} $$
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).

ag
Alan G.
Numerade Educator

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 (1 - xy^2)\ dA $. Take the sample points to be (a) the lower right corners and (b) the upper left corners of the rectangles.

ag
Alan G.
Numerade Educator

Problem 3

(a) Use a Riemann sum with $ m = n = 2 $ to estimate the value of $ \iint_R xe^{-xy}\ dA $, 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).

ag
Alan G.
Numerade Educator

Problem 4

(a) Estimate the volume of the solid that lies below the surface $ z = 1 + x^2 + 3y $ 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).

ag
Alan G.
Numerade Educator

Problem 5

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 \le x \le 4, 2 \le y \le 6 $. 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.

ag
Alan G.
Numerade Educator

Problem 6

A 20-ft-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.

ag
Alan G.
Numerade Educator

Problem 7

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)\ dA $.
(b) Estimate the average value of $ f $.

ag
Alan G.
Numerade Educator

Problem 8

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 G.
Numerade Educator

Problem 9

Evaluate the double integral by first identifying it as the volume of a solid.

$ \iint_R \sqrt{2}\ dA $, $ R = \{(x, y) \mid 2 \le x \le 6, -1 \le y \le 5 \} $

ag
Alan G.
Numerade Educator

Problem 10

Evaluate the double integral by first identifying it as the volume of a solid.

$ \iint_R (2x + 1)\ dA $, $ R = \{(x, y) \mid 0 \le x \le 2, 0 \le y \le 4 \} $

ag
Alan G.
Numerade Educator

Problem 11

Evaluate the double integral by first identifying it as the volume of a solid.

$ \iint_R (4 - 2y)\ dA $, $ R = [0, 1] \times [0, 1] $

ag
Alan G.
Numerade Educator

Problem 12

The integral $ \iint_R \sqrt{9 - y^2}\ dA $, where $ R = [0, 4] \times [0, 2] $, represents the volume of a solid. Sketch the solid.

ag
Alan G.
Numerade Educator

Problem 13

Find $ \int_0^2 f(x, y)\ dx $ and $ \int_0^3 f(x, y)\ dy $

$ f(x, y) = x + 3x^2 y^2 $

ag
Alan G.
Numerade Educator

Problem 14

Find $ \int_0^2 f(x, y)\ dx $ and $ \int_0^3 f(x, y)\ dy $

$ f(x, y) = y \sqrt{x + 2} $

ag
Alan G.
Numerade Educator

Problem 15

Calculate the iterated integral.

$ \displaystyle \int_1^4 \int_0^2 (6x^2y - 2x)\ dy dx $

ag
Alan G.
Numerade Educator

Problem 16

Calculate the iterated integral.

$ \displaystyle \int_0^1 \int_0^1 (x + y)^2\ dx dy $

ag
Alan G.
Numerade Educator

Problem 17

Calculate the iterated integral.

$ \displaystyle \int_0^1 \int_1^2 (x + e^{-y})\ dx dy $

ag
Alan G.
Numerade Educator

Problem 18

Calculate the iterated integral.

$ \displaystyle \int_0^{\frac{\pi}{6}} \int_0^{\frac{\pi}{2}} (\sin x + \sin y)\ dy dx $

ag
Alan G.
Numerade Educator

Problem 19

Calculate the iterated integral.

$ \displaystyle \int_{-3}^3 \int_0^{\frac{\pi}{2}} (y + y^2 \cos x)\ dx dy $

ag
Alan G.
Numerade Educator

Problem 20

Calculate the iterated integral.

$ \displaystyle \int_1^3 \int_1^5 \frac{\ln y}{xy}\ dy dx $

ag
Alan G.
Numerade Educator

Problem 21

Calculate the iterated integral.

$ \displaystyle \int_1^4 \int_1^2 \left (\frac{x}{y} + \frac{y}{x} \right)\ dy dx $

ag
Alan G.
Numerade Educator

Problem 22

Calculate the iterated integral.

$ \displaystyle \int_0^1 \int_0^2 ye^{x - y}\ dx dy $

ag
Alan G.
Numerade Educator

Problem 23

Calculate the iterated integral.

$ \displaystyle \int_0^3 \int_0^{\frac{\pi}{2}} t^2 \sin^3 \phi d \phi\ dt $

ag
Alan G.
Numerade Educator

Problem 24

Calculate the iterated integral.

$ \displaystyle \int_0^1 \int_0^1 xy \sqrt{x^2 + y^2}\ dy dx $

ag
Alan G.
Numerade Educator

Problem 25

Calculate the iterated integral.

$ \displaystyle \int_0^1 \int_0^1 v(u + v^2)^4\ du dv $

ag
Alan G.
Numerade Educator

Problem 26

Calculate the iterated integral.

$ \displaystyle \int_0^1 \int_0^1 \sqrt{s + t}\ ds dt $

ag
Alan G.
Numerade Educator

Problem 27

Calculate the double integral.

$ \displaystyle \iint\limits_R x \sec^2 y\ dA $, $ R = \{(x, y) \mid 0 \le x \le 2, 0 \le y \le \frac{\pi}{4} \} $

ag
Alan G.
Numerade Educator

Problem 28

Calculate the double integral.

$ \displaystyle \iint\limits_R (y + xy^{-2})\ dA $, $ R = \{(x, y) \mid 0 \le x \le 2, 1 \le y \le 2 \} $

ag
Alan G.
Numerade Educator

Problem 29

Calculate the double integral.

$ \displaystyle \iint\limits_R \frac{xy^2}{x^2 + 1}\ dA $, $ R = \{(x, y) \mid 0 \le x \le 1, -3 \le y \le 3 \} $

ag
Alan G.
Numerade Educator

Problem 30

Calculate the double integral.

$ \displaystyle \iint\limits_R \frac{\tan \theta}{\sqrt{1 - t^2}}\ dA $, $ R = \{(\theta, t) \mid 0 \le \theta \le \frac{\pi}{3}, 0 \le t \le \frac{1}{2} \} $

ag
Alan G.
Numerade Educator

Problem 31

Calculate the double integral.

$ \displaystyle \iint\limits_R x \sin (x + y)\ dA $, $ R = [0, \frac{\pi}{6}] \times [0, \frac{\pi}{3}] $

ag
Alan G.
Numerade Educator

Problem 32

Calculate the double integral.

$ \displaystyle \iint\limits_R \frac{x}{1 + xy}\ dA $, $ R = [0, 1] \times [0, 1] $

ag
Alan G.
Numerade Educator

Problem 33

Calculate the double integral.

$ \displaystyle \iint\limits_R ye^{-xy}\ dA $, $ R = [0, 2] \times [0, 3] $

ag
Alan G.
Numerade Educator

Problem 34

Calculate the double integral.

$ \displaystyle \iint\limits_R \frac{1}{1 + x + y}\ dA $, $ R = [1, 3] \times [1, 2] $

ag
Alan G.
Numerade Educator

Problem 35

Sketch the solid whose volume is given by the iterated integral.

$ \displaystyle \int_0^1 \int_0^1 (4 - x - 2y)\ dx dy $

ag
Alan G.
Numerade Educator

Problem 36

Sketch the solid whose volume is given by the iterated integral.

$ \displaystyle \int_0^1 \int_0^1 (2 - x^2 - y^2 )\ dy dx $

ag
Alan G.
Numerade Educator

Problem 37

Find the volume of the solid that lies under the plane $ 4x + 6y - 2z + 15 = 0 $ and above the rectangle $ R = \{(x, y) \mid -1 \le x \le 2, -1 \le y \le 1\} $.

ag
Alan G.
Numerade Educator

Problem 38

Find the volume of the solid that lies under the hyperbolic paraboloid $ z = 3y^2 - x^2 + 2 $ and above the rectangle $ R = [-1, 1] \times [1, 2] $.

ag
Alan G.
Numerade Educator

Problem 39

Find the volume of the solid lying under the elliptic paraboloid $ \frac{x^2}{4} + \frac{y^2}{9} + z = 1 $ and above the rectangle $ R = [-1, 1] \times [-2, 2] $.

ag
Alan G.
Numerade Educator

Problem 40

Find the volume of the solid enclosed by the surface $ z = x^2 + xy^2 $ and the planes $ z = 0 $, $ x = 0 $, $ x = 5 $, and $ y = \pm 2 $.

ag
Alan G.
Numerade Educator

Problem 41

Find the volume of the solid enclosed by the surface $ z = 1 + x^2 ye^y $ and the planes $ z = 0 $, $ x = \pm 1 $, $ y = 0 $, and $ y = 1 $.

ag
Alan G.
Numerade Educator

Problem 42

Find the volume of the solid in the first octant bounded by the cylinder $ z = 16 - x^2 $ and the plane $ y = 5 $.

ag
Alan G.
Numerade Educator

Problem 43

Find the volume of the solid enclosed by the paraboloid $ z = 2 + x^2 + (y - 2)^2 $ and the planes $ z = 1 $, $ x = 1 $, $ x = -1 $, $ y = 0 $, and $ y = 4 $.

ag
Alan G.
Numerade Educator

Problem 44

Graph the solid that lies between the surface $ z = \frac{2xy}{(x^2 + 1)} $ and the plane $ z = x + 2y $ and is bounded by the planes $ x = 0 $, $ x = 2 $, $ y = 0 $, and $ y = 4 $. Then find its volume.

ag
Alan G.
Numerade Educator

Problem 45

Use a computer algebra system to find the exact value of the integral $ \int \int_R x^5 y^3 e^{xy}\ dA $, where $ R = [0, 1] \times [0, 1] $. Then use the $ CAS $ to draw the solid whose volume is given by the integral.

ag
Alan G.
Numerade Educator

Problem 46

Graph the solid that lies between the surfaces $ z = e^{-x^2} \cos (x^2 + y^2) $ and $ z = 2 - x^2 - y^2 $ for $ \mid x \mid \le 1 $, $ \mid y \mid \le 1 $. Use a computer algebra system to approximate the volume of this solid correct to four decimal places.

ag
Alan G.
Numerade Educator

Problem 47

Find the average value of $ f $ over the given rectangle.

$ f(x, y) = x^2 y $,
$ R $ has vertices $ (-1, 0) $, $ (-1, 5) $, $ (1, 5) $, $ (1, 0) $

ag
Alan G.
Numerade Educator

Problem 48

Find the average value of $ f $ over the given rectangle.

$ f(x, y) = e^y \sqrt{x + e^y} $, $ R = [0, 4] \times [0, 1] $

ag
Alan G.
Numerade Educator

Problem 49

Use symmetry to evaluate the double integral.

$ \displaystyle \iint\limits_R \frac{xy}{1 + x^4}\ dA $, $ R = \{(x, y) \mid -1 \le x \le 1, 0 \le y \le 1 \} $

ag
Alan G.
Numerade Educator

Problem 50

Use symmetry to evaluate the double integral.

$ \displaystyle \iint\limits_R (1 + x^2 \sin y + y^2 \sin x)\ dA $, $ R = [-\pi, \pi] \times [-\pi, \pi] $

ag
Alan G.
Numerade Educator

Problem 51

Use a $ CAS $ to compute the iterated integrals

$ \displaystyle \int_0^1 \int_0^1 \frac{x - y}{(x + y)^3}\ dy dx $ and $ \displaystyle \int_0^1 \int_0^1 \frac{x - y}{(x + y)^3}\ dx dy $

Do the answers contradict Fubini's Theorem? Explain what is happening.

ag
Alan G.
Numerade Educator

Problem 52

(a) In what way are the theorems of Fubini and Clairaut similar?
(b) If $ f(x, y) $ is continuous on $ [a, b] \times [c, d] $ and
$$ g(x, y) = \int_a^x \int_c^y f(s, t)\ dt ds $$
for $ a < x < b, c < y < d $, show that $ g_{xy} = g_{yx} = f(x, y) $.

ag
Alan G.
Numerade Educator