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Chapter 7

APPLICATIONS OF INTEGRATION

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Problem 1

Find the area of the shaded region.
$$y=5 x-x^{2}$$

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Problem 2

Find the area of the shaded region.
$$y=\sqrt{x+2}$$

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Problem 3

Find the area of the shaded region.
$$x=y^{2}-2$$

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Problem 4

Find the area of the shaded region.
$$x=y^{2}-4 y$$

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Problem 5

$5-10=$ Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to $x$ or $y .$ Draw a typical
approximating rectangle and label its height and width. Then
find the area of the region.

$$y=e^{x}, \quad y=x^{2}-1, \quad x=-1, \quad x=1$$

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Problem 6

$5-10=$ Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to $x$ or $y .$ Draw a typical
approximating rectangle and label its height and width. Then
find the area of the region.

$$y=e^{x}, \quad y=x^{2}-1, \quad x=-1, \quad x=1$$

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Problem 7

$5-10=$ Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to $x$ or $y .$ Draw a typical
approximating rectangle and label its height and width. Then
find the area of the region.

$$y=(x-2)^{2}, \quad y=x$$

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Problem 8

$5-10=$ Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to $x$ or $y .$ Draw a typical
approximating rectangle and label its height and width. Then
find the area of the region.

$$y=\sin x, \quad y=2 x / \pi, \quad x \geqslant 0$$

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Problem 9

$5-10=$ Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to $x$ or $y .$ Draw a typical
approximating rectangle and label its height and width. Then
find the area of the region.

$$x=1-v^{2} \quad x=v^{2}-1$$

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Problem 10

$5-10=$ Sketch the region enclosed by the given curves. Decide
whether to integrate with respect to $x$ or $y .$ Draw a typical
approximating rectangle and label its height and width. Then
find the area of the region.

$$4 x+y^{2}=12, \quad x=y$$

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Problem 11

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$y=12-x^{2}, \quad y=x^{2}-6$$

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Problem 12

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
y=x^{2}, \quad y=4 x-x^{2}
$$

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Problem 13

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
y=e^{x}, \quad y=x e^{x}, \quad x=0
$$

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Problem 14

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
y=\cos x, \quad y=2-\cos x, \quad 0 \leqslant x \leqslant 2 \pi
$$

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Problem 15

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
x=2 y^{2}, \quad x=4+y^{2}
$$

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Problem 16

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
x=y^{4}, \quad y=\sqrt{2-x}, \quad y=0
$$

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Problem 17

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
y=\cos \pi x, \quad y=4 x^{2}-1
$$

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Problem 18

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
y=|x|, \quad y=x^{2}-2
$$

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Problem 19

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
y=1 / x, \quad y=x, \quad y=\frac{1}{4} x, \quad x>0
$$

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Problem 20

$11-20=$ Sketch the region enclosed by the given curves and
find its area.

$$
y=\frac{1}{4} x^{2}, \quad y=2 x^{2}, \quad x+y=3, \quad x \geqslant 0
$$

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Problem 21

Sketch the region that lies between the curves $y=\cos x$ and
$y=\sin 2 x$ and between $x=0$ and $x=\pi / 2 .$ Notice that the
region consists of two separate parts. Find the area of this
region.

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Problem 22

Graph the curves $y=x^{2}-x$ and $y=x^{3}-4 x^{2}+3 x$ on a
common screen and observe that the region between them
consists of two parts. Find the area of this region.

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Problem 23

$3-26=$ Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

$$y=\frac{2}{1,1, x^{4}}, \quad y=x^{2}$$

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Problem 24

$3-26=$ Graph the region between the curves and use your calculator to compute the area correct to five decimal places.
$$y=e^{1-x^{2}}, \quad y=x^{4}$$

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Problem 25

$3-26=$ Graph the region between the curves and use your calculator to compute the area correct to five decimal places.
$y=\tan ^{2} x, \quad y=\sqrt{x}$

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Problem 26

$3-26=$ Graph the region between the curves and use your calculator to compute the area correct to five decimal places.
$$y=\cos x, \quad y=x+2 \sin ^{4} x$$

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Problem 27

Racing cars driven by Chris and Kelly are side by side at
the start of a race. The table shows the velocities of each car
(in miles per hour) during the first ten seconds of the race.
Use Simpson's Rule to estimate how much farther Kelly
travels than Chris does during the first ten seconds.

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Problem 28

Two cars, $A$ and $B,$ start side by side and accelerate from
rest. The figure shows the graphs of their velocity functions.
(a) Which car is ahead after one minute? Explain.
(b) What is the meaning of the area of the shaded region?
(c) Which car is ahead after two minutes? Explain.
(d) Estimate the time at which the cars are again side by side.

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