Calculus

Gilbert Strang, Edwin “Jed” Herman

Chapter 6

Applications of Integrations - all with Video Answers

Educators

Section 1

Areas between Curves

Problem 1

Determine the area of the region between the two curves in the given figure by integrating over the $x$ -axis.
$$
y=x^{2}-3 \text { and } y=1
$$

Gregory Cho

Gregory Cho

Numerade Educator

Problem 2

Determine the area of the region between the two curves in the given figure by integrating over the $x$ -axis.
$$
y=x^{2} \text { and } y=3 x+4
$$

Carson Merrill

Numerade Educator

Problem 3

Split the region between the two curves into two smaller regions, then determine the area by integrating over the $x$ -axis. Note that you will have two integrals to solve.
$$
y=x^{3} \text { and } y=x^{2}+x
$$

Bobby Barnes

University of North Texas

Problem 4

Split the region between the two curves into two smaller regions, then determine the area by integrating over the $x$ -axis. Note that you will have two integrals to solve.
$$
y=\cos \theta \text { and } y=0.5, \text { for } 0 \leq \theta \leq \pi
$$

Amrita Bhasin

Numerade Educator

Problem 5

Determine the area of the region between the two curves by integrating over the $y$ -axis.
$$
x=y^{2} \text { and } x=9
$$

Tanishq Gupta

Numerade Educator

Problem 6

Determine the area of the region between the two curves by integrating over the $y$ -axis.
$$
y=x \text { and } x=y^{2}
$$

Gregory Cho

Numerade Educator

Problem 7

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis.
$$
y=x^{2} \text { and } y=-x^{2}+18 x
$$

Bobby Barnes

University of North Texas

Problem 8

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis.
$$
y=\frac{1}{x}, y=\frac{1}{x^{2}}, \text { and } x=3
$$

Amy Jiang

Numerade Educator

Problem 9

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis.
$$
y=\cos x \text { and } y=\cos ^{2} x \text { on } x=[-\pi, \pi]
$$

Bobby Barnes

University of North Texas

Problem 10

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis.
$$
y=e^{x}, y=e^{2 x-1}, \text { and } x=0
$$

Amy Jiang

Numerade Educator

Problem 11

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis.
$$
y=e^{x}, y=e^{-x}, x=-1 \text { and } x=1
$$

Bobby Barnes

University of North Texas

Problem 12

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis.
$$
y=e, y=e^{x}, \text { and } y=e^{-x}
$$

Amy Jiang

Numerade Educator

Problem 13

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis.
$$
y=|x| \text { and } y=x^{2}
$$

Bobby Barnes

University of North Texas

Problem 14

Graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.
$$
y=\sin (\pi x), y=2 x, \text { and } x>0
$$

Amy Jiang

Numerade Educator

Problem 15

Graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.
$$
y=12-x, y=\sqrt{x}, \text { and } y=1
$$

Bobby Barnes

University of North Texas

Problem 16

Graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.
$$
y=\sin x \text { and } y=\cos x \text { over } x=[-\pi, \pi]
$$

Carson Merrill

Numerade Educator

Problem 17

Graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.
$$
y=x^{3} \text { and } y=x^{2}-2 x \text { over } x=[-1,1]
$$

Bobby Barnes

University of North Texas

Problem 18

Graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.
$$
y=x^{2}+9 \text { and } y=10+2 x \text { over } x=[-1,3]
$$

Amy Jiang

Numerade Educator

Problem 19

Graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.
$$
y=x^{3}+3 x \text { and } y=4 x
$$

Bobby Barnes

University of North Texas

Problem 20

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y$ -axis.
$$
x=y^{3} \text { and } x=3 y-2
$$

Amy Jiang

Numerade Educator

Problem 21

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y$ -axis.
$$
x=2 y \text { and } x=y^{3}-y
$$

Amy Jiang

Numerade Educator

Problem 22

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y$ -axis.
$$
y^{2}=x \text { and } x=y+2
$$

Bobby Barnes

University of North Texas

Problem 23

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y$ -axis.
$$
x=|y| \text { and } 2 x=-y^{2}+2
$$

Amy Jiang

Numerade Educator

Problem 24

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y$ -axis.
$$
x=\sin y, x=\cos (2 y), y=\pi / 2, \text { and } y=-\pi / 2
$$

Bobby Barnes

University of North Texas

Problem 25

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
x=y^{4} \text { and } x=y^{5}
$$

Amy Jiang

Numerade Educator

Problem 26

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=x e^{x}, y=e^{x}, x=0, \text { and } x=1
$$

Bobby Barnes

University of North Texas

Problem 27

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=x^{6} \text { and } y=x^{4}
$$

Amy Jiang

Numerade Educator

Problem 28

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
x=y^{3}+2 y^{2}+1 \text { and } x=-y^{2}+1
$$

Bobby Barnes

University of North Texas

Problem 29

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=|x| \text { and } y=x^{2}-1
$$

Amy Jiang

Numerade Educator

Problem 30

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=4-3 x \text { and } y=\frac{1}{x}
$$

Bobby Barnes

University of North Texas

Problem 31

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=\sin x, x=-\pi / 6, x=\pi / 6, \text { and } y=\cos ^{3} x
$$

Amy Jiang

Numerade Educator

Problem 32

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=x^{2}-3 x+2 \text { and } y=x^{3}-2 x^{2}-x+2
$$

Bobby Barnes

University of North Texas

Problem 33

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=2 \cos ^{3}(3 x), y=-1, x=\frac{\pi}{4}, \text { and } x=-\frac{\pi}{4}
$$

Amrita Bhasin

Numerade Educator

Problem 34

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y+y^{3}=x \text { and } 2 y=x
$$

Bobby Barnes

University of North Texas

Problem 35

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=\sqrt{1-x^{2}} \text { and } y=x^{2}-1
$$

Amy Jiang

Numerade Educator

Problem 36

Graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$ -axis or $y$ -axis, whichever seems more convenient.
$$
y=\cos ^{-1} x, y=\sin ^{-1} x, x=-1, \text { and } x=1
$$

Taylor Shimono

Numerade Educator

Problem 37

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [? } x=e^{y} \text { and } y=x-2
$$

Amy Jiang

Numerade Educator

Problem 38

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [T] } y=x^{2} \text { and } y=\sqrt{1-x^{2}}
$$

Taylor Shimono

Numerade Educator

Problem 39

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [7] } y=3 x^{2}+8 x+9 \text { and } 3 y=x+24
$$

Amy Jiang

Numerade Educator

Problem 40

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [T] } x=\sqrt{4-y^{2}} \text { and } y^{2}=1+x^{2}
$$

Taylor Shimono

Numerade Educator

Problem 41

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [T] } x^{2}=y^{3} \text { and } x=3 y
$$

Amrita Bhasin

Numerade Educator

Problem 42

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [T] } y=\sin ^{3} x+2, y=\tan x, x=-1.5, \text { and } x=1.5
$$

Taylor Shimono

Numerade Educator

Problem 43

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [? } y=\sqrt{1-x^{2}} \text { and } y^{2}=x^{2}
$$

Amy Jiang

Numerade Educator

Problem 44

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [T] } y=\sqrt{1-x^{2}} \text { and } y=x^{2}+2 x+1
$$

Bobby Barnes

University of North Texas

Problem 45

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [T] } x=4-y^{2} \text { and } x=1+3 y+y^{2}
$$

Amrita Bhasin

Numerade Educator

Problem 46

Find the exact area of the region bounded by the given equations if possible. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region.
$$
\text { [? } y=\cos x, y=e^{x}, x=-\pi \text { , and } x=0
$$

Bobby Barnes

University of North Texas

Problem 47

A factory selling cell phones has a marginal cost function $C(x)=0.01 x^{2}-3 x+229$, where $x$ represents the number of cell phones, and a marginal revenue function given by $R(x)=429-2 x$. Find the area between the graphs of these curves and $x=0 .$ What does this area represent?

Bobby Barnes

University of North Texas

Problem 48

An amusement park has a marginal cost function $C(x)=1000 e^{-x}+5$, where $x$ represents the number of tickets sold, and a marginal revenue function given by $R(x)=60-0.1 x$. Find the total profit generated when selling 550 tickets. Use a calculator to determine intersection points, if necessary, to two decimal places.

Amy Jiang

Numerade Educator

Problem 49

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H(t)=1-\cos ((\pi t) / 2)$ whereas the speed of the tortoise is $T(t)=(1 / 2) \tan ^{-1}(t / 4)$, where $t$ is time measured in hours and the speed is measured in miles per hour. Find the area between the curves from time $t=0$ to the first time after one hour when the tortoise and hare are traveling at the same speed. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

Taylor Shimono

Numerade Educator

Problem 50

The tortoise versus the hare: The speed of the hare is given by the sinusoidal function $H(t)=(1 / 2)-(1 / 2) \cos (2 \pi t)$ whereas the speed of the tortoise is $T(t)=\sqrt{t}$, where $t$ is time measured in hours and speed is measured in kilometers per hour. If the race is over in 1 hour, who won the race and by how much? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

Amy Jiang

Numerade Educator

Problem 51

Find the area between the curves by integrating with respect to $x$ and then with respect to $y .$ Is one method easier than the other? Do you obtain the same answer?
$$
y=x^{2}+2 x+1 \text { and } y=-x^{2}-3 x+4
$$

Taylor Shimono

Numerade Educator

Problem 52

Find the area between the curves by integrating with respect to $x$ and then with respect to $y .$ Is one method easier than the other? Do you obtain the same answer?
$$
y=x^{4} \text { and } x=y^{5}
$$

Amrita Bhasin

Numerade Educator

Problem 53

Find the area between the curves by integrating with respect to $x$ and then with respect to $y .$ Is one method easier than the other? Do you obtain the same answer?
$$
x=y^{2}-2 \text { and } x=2 y
$$

Taylor Shimono

Numerade Educator

Problem 54

Determine the equations for the sides of the square that touches the unit circle on all four sides, as seen in the following figure. Find the area between the perimeter of this square and the unit circle. Is there another way to solve this without using calculus?

Amy Jiang

Numerade Educator

Problem 55

Find the area between the perimeter of the unit circle and the triangle created from $y=2 x+1, y=1-2 x$ and $y=-\frac{3}{5}$, as seen in the following figure. Is there a way to solve this without using calculus?

Taylor Shimono

Numerade Educator