Calculus Early Transcendentals

Howard Anton, Irl Bivens, Stephen Davis

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING - all with Video Answers

Educators

Section 1

Area Between Two Curves

Problem 1

Find the area of the shaded region.
(Graph cant copy)

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Problem 2

Find the area of the shaded region.
(Graph cant copy)

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Problem 3

Find the area of the shaded region.
(Graph cant copy)

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Problem 4

Find the area of the shaded region.
(Graph cant copy)

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Problem 5

Find the area of the shaded region by (a) integrating with
respect to x and (b) integrating with respect to y.
(Graph cant copy)

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Problem 6

Find the area of the shaded region by (a) integrating with
respect to x and (b) integrating with respect to y.
(Graph cant copy)

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Problem 7

Sketch the region enclosed by the curves and find its area.
$$
y=x^{2}, y=\sqrt{x}, x=\frac{1}{4}, x=1
$$

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Problem 8

Sketch the region enclosed by the curves and find its area.
$$
y=x^{3}-4 x, y=0, x=0, x=2
$$

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Problem 9

Sketch the region enclosed by the curves and find its area.
$$
y=\cos 2 x, y=0, x=\pi / 4, x=\pi / 2
$$

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Problem 10

Sketch the region enclosed by the curves and find its area.
$$
y=\sec ^{2} x, y=2, x=-\pi / 4, x=\pi / 4
$$

Gregory H.

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Problem 11

Sketch the region enclosed by the curves and find its area.
$$
x=\sin y, x=0, y=\pi / 4, y=3 \pi / 4
$$

Gregory H.

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Problem 12

Sketch the region enclosed by the curves and find its area.
$$
x^{2}=y, x=y-2
$$

Gregory H.

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Problem 13

Sketch the region enclosed by the curves and find its area.
$$
y=e^{x}, y=e^{2 x}, x=0, x=\ln 2
$$

Gregory H.

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Problem 14

Sketch the region enclosed by the curves and find its area.
$$
x=1 / y, x=0, y=1, y=e
$$

Gregory H.

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Problem 15

Sketch the region enclosed by the curves and find its area.
$$
y=\frac{2}{1+x^{2}}, y=|x|
$$

Gregory H.

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Problem 16

Sketch the region enclosed by the curves and find its area.
$$
y=\frac{1}{\sqrt{1-x^{2}}}, y=2
$$

Gregory H.

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Problem 17

Sketch the region enclosed by the curves and find its area.
$$
y=2+|x-1|, \quad y=-\frac{1}{5} x+7
$$

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Problem 18

Sketch the region enclosed by the curves and find its area.
$$
y=x, y=4 x, y=-x+2
$$

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Problem 19

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=x^{3}-4 x^{2}+3 x, \quad y=0
$$

Gregory H.

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Problem 20

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=x^{3}-2 x^{2}, y=2 x^{2}-3 x
$$

Gregory H.

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Problem 21

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=\sin x, y=\cos x, x=0, x=2 \pi
$$

Gregory H.

Numerade Educator

Problem 22

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=x^{3}-4 x, y=0
$$

Gregory H.

Numerade Educator

Problem 23

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
x=y^{3}-y, x=0
$$

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Problem 24

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
x=y^{3}-4 y^{2}+3 y, x=y^{2}-y
$$

Gregory H.

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Problem 25

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=x e^{x^{2}}, y=2|x|
$$

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Problem 26

Use a graphing utility, where helpful, to find the area of the region enclosed by the curves.
$$
y=\frac{1}{x \sqrt{1-(\ln x)^{2}}}, y=\frac{3}{x}
$$

Gregory H.

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Problem 27

True-False Determine whether the statement is true or false. Explain your answer. [In each exercise, assume that $f$
and $g$ are distinct continuous functions on $[a, b]$ and that $A$ de-
notes the area of the region bounded by the graphs of $y=f(x),$
$y=g(x), x=a, \text { and } x=b .]$
$$
\text { If } f \text { and } g \text { differ by a positive constant } c, \text { then } A=c(b-a)
$$

Gregory H.

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Problem 28

True-False Determine whether the statement is true or false. Explain your answer. [In each exercise, assume that $f$
and $g$ are distinct continuous functions on $[a, b]$ and that $A$ de-
notes the area of the region bounded by the graphs of $y=f(x),$
$y=g(x), x=a, \text { and } x=b .]$

If
$$
\int_{a}^{b}[f(x)-g(x)] d x=-3
$$
$$
\text { then } A=3
$$

Gregory H.

Numerade Educator

Problem 29

True-False Determine whether the statement is true or false. Explain your answer. [In each exercise, assume that $f$
and $g$ are distinct continuous functions on $[a, b]$ and that $A$ de-
notes the area of the region bounded by the graphs of $y=f(x),$
$y=g(x), x=a, \text { and } x=b .]$

$$
\text { If } \quad \int_{a}^{b}[f(x)-g(x)] d x=0
$$
$$
\begin{array}{l}{\text { then the graphs of } y=f(x) \text { and } y=g(x) \text { cross at least }} \\ {\text { once on }[a, b] .}\end{array}
$$

Gregory H.

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Problem 30

True-False Determine whether the statement is true or false. Explain your answer. [In each exercise, assume that $f$
and $g$ are distinct continuous functions on $[a, b]$ and that $A$ de-
notes the area of the region bounded by the graphs of $y=f(x),$
$y=g(x), x=a, \text { and } x=b .]$

$$
\text { If } \quad A=\left|\int_{a}^{b}[f(x)-g(x)] d x\right|
$$
$$
\begin{array}{l}{\text { then the graphs of } y=f(x) \text { and } y=g(x) \text { don't cross on }} \\ {[a, b] .}\end{array}
$$

Gregory H.

Numerade Educator

Problem 31

Estimate the value of $k(0<k<1)$ so that the region enclosed by $y=1 / \sqrt{1-x^{2}}, y=x, x=0,$ and $x=k$ has an
area of 1 square unit.

Gregory H.

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Problem 32

Estimate the area of the region in the first quadrant enclosed
by $y=\sin 2 x$ and $y=\sin ^{-1} x$

Gregory H.

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Problem 33

Use a CAS to find the area enclosed by $y=3-2 x$ and
$y=x^{6}+2 x^{5}-3 x^{4}+x^{2}$

Gregory H.

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Problem 34

Use a CAS to find the exact area enclosed by the curves
$y=x^{5}-2 x^{3}-3 x$ and $y=x^{3}$.

Gregory H.

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Problem 35

Find a horizontal line $y=k$ that divides the area between
$y=x^{2}$ and $y=9$ into two cqual parts.

Gregory H.

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Problem 36

Find a vertical line $x=k$ that divides the area enclosed by
$x=\sqrt{y}, x=2,$ and $y=0$ into two equal parts.

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Problem 37

(a) Find the area of the region enclosed by the parabola
$y=2 x-x^{2}$ and the $x$ -axis.
(b) Find the value of $m$ so that the line $y=m x$ divides the
region in part (a) into two regions of equal area.

Gregory H.

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Problem 38

Find the area between the curve $y=\sin x$ and the line segment joining the points $(0,0)$ and $(5 \pi / 6,1 / 2)$ on the curve.

Gregory H.

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Problem 39

Use Newton's Method (Section 4.7), where needed, to approximate the $x$ -coordinates of the intersections of the curves to at least four decimal places, and then use those approximations
to approximate the area of the region.
$$
\begin{array}{l}{\text { The region that lies below the curve } y=\sin x \text { and above }} \\ {\text { the line } y=0.2 x, \text { where } x \geq 0}\end{array}
$$

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Problem 40

Use Newton's Method (Section 4.7), where needed, to approximate the $x$ -coordinates of the intersections of the curves to at least four decimal places, and then use those approximations
to approximate the area of the region.
$$
\text { The region enclosed by the graphs of } y=x^{2} \text { and } y=\cos x
$$

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Problem 41

Use Newton's Method (Section 4.7), where needed, to approximate the $x$ -coordinates of the intersections of the curves to at least four decimal places, and then use those approximations
to approximate the area of the region.

$$
\text { The region enclosed by the graphs of } y=(\ln x) / x \text { and }
$$

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Problem 42

Use Newton's Method (Section 4.7), where needed, to approximate the $x$ -coordinates of the intersections of the curves to at least four decimal places, and then use those approximations
to approximate the area of the region.
$$
\begin{array}{l}{\text { The region enclosed by the graphs of } y=3-2 \cos x \text { and }} \\ {y=2 /\left(1+x^{2}\right)}\end{array}
$$

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Problem 43

Use Newton's Method (Section 4.7), where needed, to approximate the $x$ -coordinates of the intersections of the curves to at least four decimal places, and then use those approximations
to approximate the area of the region.
$$
\begin{array}{l}{\text { The region enclosed by the graphs of } y=x^{2}-1 \text { and }} \\ {y=2 \sin x .}\end{array}
$$

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Problem 44

Referring to the accompanying figure, use a CAS to esti-
mate the value of $k$ so that the areas of the shaded regions
are equal.
$$
\begin{array}{l}{\text { Source: This exercise is bused on } \mathrm{P} \text { poblem } \mathrm{Al} \text { that was posed in the Fifty-Fourth }} \\ {\text { Annual William Lowell Putnam Mathematical Competition. }}\end{array}
$$

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Problem 45

Two racers in adjacent lanes move with velocity functions $v_{1}(t) \mathrm{m} / \mathrm{s}$ and $v_{2}(t) \mathrm{m} / \mathrm{s}$, respectively. Suppose
that the racers are even at time $t=60 \mathrm{s}$. Interpret the value of the integral
$$
\int_{0}^{60}\left[v_{2}(t)-v_{1}(t)\right] d t
$$
in this context.

Gregory H.

Numerade Educator

Problem 46

The accompanying figure shows acceleration versus
time curves for two cars that move along a straight track,
accelerating from rest at the starting line. What does the
area $A$ between the curves over the interval $0 \leq t \leq T$
represent? Justify your answer.

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Problem 47

Suppose that $f$ and $g$ are integrable on $[a, b],$ but neither
$f(x) \geq g(x)$ nor $g(x) \geq f(x)$ holds for all $x$ in $[a, b]$
[i.e., the curves $y=f(x)$ and $y=g(x)$ are intertwined].
(a) What is the geometric significance of the integral
$$
\int_{a}^{b}[f(x)-g(x)] d x ?
$$
$$
\text { (b) What is the geometric significance of the integral }
$$
$$
\int_{a}^{b}|f(x)-g(x)| d x ?
$$

Gregory H.

Numerade Educator

Problem 48

Let $A(n)$ be the area in the first quadrant enclosed by
the curves $y=\sqrt[n]{x}$ and $y=x$
(a) By considering how the graph of $y=\sqrt[n]{x}$ changes
as $n$ increases, make a conjecture about the limit of
$A(n)$ as $n \rightarrow+\infty$.
(b) Confirm your conjecture by calculating the limit.

Gregory H.

Numerade Educator

Problem 49

Find the area of the region enclosed between the curve
$x^{1 / 2}+y^{1 / 2}=a^{1 / 2}$ and the coordinate axes.

Gregory H.

Numerade Educator

Problem 50

Show that the area of the ellipse in the accompanying figure
is $\pi a b .[\text {Hint} \text { : Use a formula from geometry. }]$

Carson M.

Numerade Educator

Problem 51

Writing Suppose that $f$ and $g$ are continuous on $[a, b]$
but that the graphs of $y=f(x)$ and $y=g(x)$ cross several times. Describe a step-by-step procedure for determining the area bounded by the graphs of $y=f(x), y=g(x)$
$x=a,$ and $x=b$

Gregory H.

Numerade Educator

Problem 52

Writing Suppose that $R$ and $S$ are two regions in the $x y$ plane that lie between a pair of lines $L_{1}$ and $L_{2}$ that are parallel to the $y$ -axis. Assume that each line between $L_{1}$
and $L_{2}$ that is parallel to the $y$ -axis intersects $R$ and $S$ in line segments of equal length. Give an informal argument
that the area of $R$ is equal to the area of $S$. (Make reasonable
assumptions about the boundaries of $R$ and $S$.)

Gregory H.

Numerade Educator