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Chapter 6

INTEGRALS

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Problem 1

$1-4$ Find the area of the shaded region.

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Problem 2

$1-4$ Find the area of the shaded region.

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Problem 3

$1-4$ Find the area of the shaded region.

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Problem 4

$1-4$ Find the area of the shaded region.

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Problem 5

$5-28$ 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.

$y=x+1, \quad y=9-x^{2}, \quad x=-1, \quad x=2$

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Problem 6

$5-28$ 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.

$y=\sin x, \quad y=e^{x}, \quad x=0, \quad x=\pi / 2$

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Problem 7

$5-28$ 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.

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

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Problem 8

$5-28$ 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.

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

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Problem 9

$5-28$ 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.


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

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Problem 10

$5-28$ 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.


$y=1+\sqrt{x}, \quad y=(3+x) / 3$

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Problem 11

$5-28$ 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.


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

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Problem 12

$5-28$ 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.


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

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Problem 13

$5-28$ 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.


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

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Problem 14

$5-28$ 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.


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

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Problem 15

$5-28$ 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.


$y=\tan x, \quad y=2 \sin x, \quad-\pi / 3 \leqslant x \leqslant \pi / 3$

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Problem 16

$5-28$ 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.


$y=x^{3}-x, \quad y=3 x$

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Problem 17

$5-28$ 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.


$$y=\sqrt{x}, \quad y=\frac{1}{2} x, \quad x=9$$

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Problem 18

$5-28$ 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.


$$y=8-x^{2}, \quad y=x^{2}, \quad x=-3, \quad x=3$$

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Problem 19

$5-28$ 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.


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

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Problem 20

$5-28$ 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 21

$5-28$ 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.


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

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Problem 22

$5-28$ 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.


$$y=\sin (\pi x / 2), \quad y=x$$

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Problem 23

$5-28$ 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.


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

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Problem 24

$5-28$ 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.


$$y=\cos x, \quad y=1-\cos x, \quad 0 \leqslant x \leqslant \pi$$

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Problem 25

$5-28$ 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.


$$y=x^{2}, \quad y=2 /\left(x^{2}+1\right)$$

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Problem 26

$5-28$ 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.


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

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Problem 27

$5-28$ 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.


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

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Problem 28

$5-28$ 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.


$$y=3 x^{2}, \quad y=8 x^{2}, \quad 4 x+y=4, \quad x \geqslant 0$$

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Problem 29

$29-30$ Use calculus to find the area of the triangle with the given
vertices.

$$(0,0), \quad(2,1), \quad(-1,6)$$

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Problem 30

$29-30$ Use calculus to find the area of the triangle with the given
vertices.

$$(0,5), \quad(2,-2), \quad(5,1)$$

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Problem 31

$31-32$ Evaluate the integral and interpret it as the area of a region. Sketch the region.

$$\int_{0}^{\pi / 2}|\sin x-\cos 2 x| d x$$

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Problem 32

$31-32$ Evaluate the integral and interpret it as the area of a region. Sketch the region.

$$\int_{0}^{4}|\sqrt{x+2}-x| d x$$

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Problem 33

$33-34$ Use the Midpoint Rule with $n=4$ to approximate the area of the region bounded by the given curves.

$$y=\sin ^{2}(\pi x / 4), \quad v=\cos ^{2}(\pi x / 4), \quad 0 \leq x \leqslant 1$$

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Problem 34

$33-34$ Use the Midpoint Rule with $n=4$ to approximate the area of the region bounded by the given curves.

$$y=\sqrt[3]{16-x^{3}}, \quad y=x, \quad x=0$$

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Problem 35

$35-38$ Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$$y=x \sin \left(x^{2}\right), \quad y=x^{4}$$

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Problem 36

$35-38$ Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

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

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Problem 37

$35-38$ Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$$y=3 x^{2}-2 x, \quad y=x^{3}-3 x+4$$

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Problem 38

$35-38$ Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$y=x \cos x, \quad y=x^{10}$$

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Problem 39

Use a computer algebra system to find the exact area enclosed by the curves $y=x^{5}-6 x^{3}+4 x$ and $y=x$ .

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Problem 40

Sketch the region in the $x y$ -plane defined by the inequalities $$x-2 y^{2} \geqslant 0,1-x-|y| \geqslant 0$$ and find its area.

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Problem 41

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 the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

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Problem 42

The widths (in meters) of a kidney-shaped swimming pool were measured at 2 -meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.

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Problem 43

A cross-section of an airplane wing is shown. Measurements of the height of the wing, in centimeters, at 20 -centimeter intervals are $5.8,20.3,26.7,29.0,27.6,27.3,23.8,20.5,15.1$
$8.7,$ and $2.8 .$ Use the Midpoint Rule to estimate the area of the wing's cross-section.

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Problem 44

If the birth rate of a population is $b(t)=2200 e^{0.024 t}$ people per year and the death rate is $d(t)=1460 e^{0.0218 t}$ people per year, find the area between these curves for 0$\leqslant t \leqslant 10 .$ What does this area represent?

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Problem 45

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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Problem 46

The figure shows graphs of the marginal revenue function $R^{\prime}$ and the marginal cost function $C^{\prime}$ for a manufacturer. [Recall from Section 4.7 that $R(x)$ and $C(x)$ represent the revenue and cost when $x$ units are manufactured. Assume that $R$ and $C$ are measured in thousands of dollars. J What is the meaning of the
area of the shaded region? Use the Midpoint Rule to estimate the value of this quantity.

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Problem 47

The curve with equation $y^{2}=x^{2}(x+3)$ is called Tschirnhausen's cubic. If you graph this curve you will see that part
of the curve forms a loop. Find the area enclosed by the loop.

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Problem 48

Find the area of the region bounded by the parabola $y=x^{2}$
the tangent line to this parabola at $(1,1),$ and the $x$ -axis.

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Problem 49

Find the number $b$ such that the line $y=b$ divides the region
bounded by the curves $y=x^{2}$ and $y=4$ into two regions
with equal area.

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Problem 50

(a) Find the number a such that the line $x=a$ bisects the
area under the curve $y=1 / x^{2}, 1 \leqslant x \leqslant 4 .$
(b) Find the number $b$ such that the line $y=b$ bisects the
area in part (a).

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Problem 51

Find the values of $c$ such that the area of the region bounded
by the parabolas $y=x^{2}-c^{2}$ and $y=c^{2}-x^{2}$ is $576 .$

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Problem 52

Suppose that $0<c<\pi / 2 .$ For what value of $c$ is the area of
the region enclosed by the curves $y=\cos x, y=\cos (x-c)$
and $x=0$ equal to the area of the region enclosed by the
curves $v=\cos (x-c), x=\pi$ and $V=0 ?$

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Problem 53

For what values of $m$ do the line $y=m x$ and the curve
$y=x /\left(x^{2}+1\right)$ enclose a region? Find the area of the region.

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