Calculus Single Variable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

Chapter 8

Using the Definite Integral - all with Video Answers

Educators

Section 1

Areas and Volumes

Problem 1

(a) Write a Riemann sum approximating the area of the region in Figure $8.13,$ using vertical strips as shown.
(b) Evaluate the corresponding integral.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 2

(a) Write a Riemann sum approximating the area of the region in Figure $8.14,$ using vertical strips as shown.
(b) Evaluate the corresponding integral.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 3

(a) Write a Riemann sum approximating the area of the region in Figure $8.15,$ using horizontal strips as shown.
(b) Evaluate the corresponding integral.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 4

(a) Write a Riemann sum approximating the area of the region in Figure $8.16,$ using horizontal strips as shown.
(b) Evaluate the corresponding integral.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 5

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 6

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 7

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 8

Write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 9

In Exercises $5-12,$ write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Lucía Guerrero

Numerade Educator

Problem 10

In Exercises $5-12,$ write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 11

In Exercises $5-12,$ write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 12

In Exercises $5-12,$ write a Riemann sum and then a definite integral representing the area of the region, using the strip shown. Evaluate the integral exactly.

Lucía Guerrero

Numerade Educator

Problem 13

(a) Match the regions I-IV in Figure 8.17 with the regions $A-D:$
- A: Bounded by $y+x=2, y=x, x=2$
- Bounded by $y+x=2, y=x, y=2$
- $C:$ Bounded by $y+x=2, y=x, x=0$
$D:$ Bounded by $y+x=2, y=x, y=0$
(b) Write integrals representing the areas of the regions II and III using vertical strips. Do not evaluate.

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 14

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 15

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 16

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 17

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 18

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 19

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 20

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 21

Write a Ricmann sum and then a definite integral representing the volume of the region, using the slice shown. Evaluate the integral exactly. (Regions are parts of cones, cylinders, spheres, pyramids, and triangular prisms.)

Cooper Wilkinson

Cooper Wilkinson

Numerade Educator

Problem 22

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the base and height of the triangle or the radius of the circle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{1} 3 x d x$$

Gregory Higby

Numerade Educator

Problem 23

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the base and height of the triangle or the radius of the circle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{-9}^{9} \sqrt{81-x^{2}} d x$$

Gregory Higby

Numerade Educator

Problem 24

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the base and height of the triangle or the radius of the circle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{\sqrt{15}} \sqrt{15-h^{2}} d h$$

Gregory Higby

Numerade Educator

Problem 25

Represent the area of either a triangle or part of a circle, and the variable of integration measures a distance. In each case, say which shape is represented, and give the base and height of the triangle or the radius of the circle. Make a sketch to support your answer showing the variable and all other relevant quantities.
$$\int_{0}^{7} 5\left(1-\frac{h}{7}\right) d h$$

Gregory Higby

Numerade Educator

Problem 26

The integral $\int_{0}^{1}\left(x-x^{2}\right) d x$ represents the area of a region between two curves in the plane. Make a sketch of this region.

Gregory Higby

Numerade Educator

Problem 27

Construct and evaluate definite integral(s) representing the area of the region described, using:
(a) Vertical slices
(b) Horizontal slices
Enclosed by $y=x^{2}$ and $y=3 x$

Gregory Higby

Numerade Educator

Problem 28

Construct and evaluate definite integral(s) representing the area of the region described, using:
(a) Vertical slices
(b) Horizontal slices
Enclosed by $y=2 x$ and $y=12-x$ and the $y$ -axis.

Gregory Higby

Numerade Educator

Problem 29

Construct and evaluate definite integral(s) representing the area of the region described, using:
(a) Vertical slices
(b) Horizontal slices
Enclosed by $y=x^{2}$ and $y=6-x$ and the $x$ -axis.

Gregory Higby

Numerade Educator

Problem 30

Construct and evaluate definite integral(s) representing the area of the region described, using:
(a) Vertical slices
(b) Horizontal slices
Enclosed by $y=2 x$ and $x=5$ and $y=6$ and the $x$ -axis.

Gregory Higby

Numerade Educator

Problem 31

The integral represents the volume of a hemisphere, sphere, or cone, and the variable of integration is a length. Say which shape is represented; give the radius of the hemisphere or sphere or the radius and height of the cone. Make a sketch showing the variable and all relevant quantities.
$$\int_{0}^{12} \pi\left(144-h^{2}\right) d h$$

Gregory Higby

Numerade Educator

Problem 32

The integral represents the volume of a hemisphere, sphere, or cone, and the variable of integration is a length. Say which shape is represented; give the radius of the hemisphere or sphere or the radius and height of the cone. Make a sketch showing the variable and all relevant quantities.
$$\int_{0}^{12} \pi(x / 3)^{2} d x$$

Gregory Higby

Numerade Educator

Problem 33

The integral represents the volume of a hemisphere, sphere, or cone, and the variable of integration is a length. Say which shape is represented; give the radius of the hemisphere or sphere or the radius and height of the cone. Make a sketch showing the variable and all relevant quantities.
$$2 \int_{0}^{8} \pi\left(64-h^{2}\right) d h$$

Gregory Higby

Numerade Educator

Problem 34

The integral represents the volume of a hemisphere, sphere, or cone, and the variable of integration is a length. Say which shape is represented; give the radius of the hemisphere or sphere or the radius and height of the cone. Make a sketch showing the variable and all relevant quantities.
$$\int_{0}^{6} \pi(3-y / 2)^{2} d y$$

Gregory Higby

Numerade Educator

Problem 35

The integral represents the volume of a hemisphere, sphere, or cone, and the variable of integration is a length. Say which shape is represented; give the radius of the hemisphere or sphere or the radius and height of the cone. Make a sketch showing the variable and all relevant quantities.
$$\int_{0}^{2} \pi\left(2^{2}-(2-y)^{2}\right) d y$$

Gregory Higby

Numerade Educator

Problem 36

A cone with base radius 4 and height 16 standing with its vertex upward is cut into 32 horizontal slices of equal thickness $\Delta h$
(a) Find $\Delta h$
(b) Find a formula relating the radius of the cone $r$ at height $h$ above the base.
(c) What is the approximate volume of the bottom slice?
(d) What is the height of the ninth slice from the bottom?
(e) What is the approximate volume of the ninth slice?

Cory Kuzinski

Numerade Educator

Problem 37

A hemisphere with a horizontal base of radius 4 is cut parallel to the base into 20 slices of equal thickness.
(a) What is the thickness $\Delta h$ of each slice?
(b) What is the approximate volume of the slice at height $h$ above the base?

Himanshu Kushwaha

Himanshu Kushwaha

Numerade Educator

Problem 38

Find the volume of a sphere of radius $r$ by slicing.

Gregory Higby

Numerade Educator

Problem 39

Set up and evaluate an integral to find the volume of a cone of height $12 \mathrm{m}$ and base radius $3 \mathrm{m}$.

Nick Johnson

Numerade Educator

Problem 40

Find, by slicing, a formula for the volume of a cone of height $h$ and base radius $r .$

Gregory Higby

Numerade Educator

Problem 41

Figure 8.18 shows a solid with both rectangular and triangular cross sections.
(a) Slice the solid parallel to the triangular faces. Sketch one slice and calculate its volume in terms of $x,$ the distance of the slice from one end. Then
Write and evaluate an integral giving the volume of the solid.
(b) Repeat part (a) for horizontal slices. Instead of $x$, use $h,$ the distance of a slice from the top.

Amy Jiang

Numerade Educator

Problem 42

A rectangular lake is $150 \mathrm{km}$ long and $3 \mathrm{km}$ wide. The vertical cross-section through the lake in Figure 8.19 shows that the lake is $0.2 \mathrm{km}$ deep at the center. (These are the approximate dimensions of Lake Mead, a large reservoir providing water to California, Nevada, and Arizona.) Set up and evaluate a definite integral giving the total volume of water in the lake.

Bobby Barnes

University of North Texas

Problem 43

A dam has a rectangular base 1400 meters long and 160 meters wide. Its cross-section is shown in Figure $8.20 .$ (The Grand Coulee Dam in Washington state is roughly this size.) By slicing horizontally, set up and evaluate a definite integral giving the volume of material used to build this dam.

Tanishq Gupta

Numerade Educator

Problem 44

(a) Set up and evaluate an integral giving the volume of a pyramid of height $10 \mathrm{m}$ and square base $8 \mathrm{m}$ by $8 \mathrm{m}$
(b) The pyramid in part (a) is cut off at a height of 6
m. See Figure $8.21 .$ Find the volume.

Dorcas Attuabea Addo

Dorcas Attuabea Addo

Numerade Educator

Problem 45

The exterior of a holding tank is a cylinder with radius $3 \mathrm{m}$ and height $6 \mathrm{m} ;$ the interior is cone-shaped; Figure 8.22 shows its cross-section. Using an integral, find the volume of material needed to make the tank.

Gregory Higby

Numerade Educator

Problem 46

The given volume has a horizontal base. Let $h$ be the height above the base of a slice with thickness $\Delta h .$ Which of $(\mathrm{I})-(\mathrm{IV})$ approximates the volume of this slice?
I. $\pi \sqrt{16-h^{2}} \Delta h$
II. $\quad \frac{\pi}{25}(20-h)^{2} \Delta h$
III. $25 \pi(20-h)^{2} \Delta h$
IV. $\quad \pi\left(16-h^{2}\right) \Delta h$
V. $\frac{1}{25}(20-h)^{2} \Delta h$
VI. $25(20-h)^{2} \Delta h$
A cone of height $h=20$ and base radius $r=4$

Donald Yeh

Numerade Educator

Problem 47

The given volume has a horizontal base. Let $h$ be the height above the base of a slice with thickness $\Delta h .$ Which of $(\mathrm{I})-(\mathrm{IV})$ approximates the volume of this slice?
I. $\pi \sqrt{16-h^{2}} \Delta h$
II. $\quad \frac{\pi}{25}(20-h)^{2} \Delta h$
III. $25 \pi(20-h)^{2} \Delta h$
IV. $\quad \pi\left(16-h^{2}\right) \Delta h$
V. $\frac{1}{25}(20-h)^{2} \Delta h$
VI. $25(20-h)^{2} \Delta h$
A cone of height $h=20$ and base radius $r=100$.

Donald Yeh

Numerade Educator

Problem 48

The given volume has a horizontal base. Let $h$ be the height above the base of a slice with thickness $\Delta h .$ Which of $(\mathrm{I})-(\mathrm{IV})$ approximates the volume of this slice?
I. $\pi \sqrt{16-h^{2}} \Delta h$
II. $\quad \frac{\pi}{25}(20-h)^{2} \Delta h$
III. $25 \pi(20-h)^{2} \Delta h$
IV. $\quad \pi\left(16-h^{2}\right) \Delta h$
V. $\frac{1}{25}(20-h)^{2} \Delta h$
VI. $25(20-h)^{2} \Delta h$
A pyramid whose base is a square of side $s=4$ and whose height is $h=20$.

Donald Yeh

Numerade Educator

Problem 49

The given volume has a horizontal base. Let $h$ be the height above the base of a slice with thickness $\Delta h .$ Which of $(\mathrm{I})-(\mathrm{IV})$ approximates the volume of this slice?
I. $\pi \sqrt{16-h^{2}} \Delta h$
II. $\quad \frac{\pi}{25}(20-h)^{2} \Delta h$
III. $25 \pi(20-h)^{2} \Delta h$
IV. $\quad \pi\left(16-h^{2}\right) \Delta h$
V. $\frac{1}{25}(20-h)^{2} \Delta h$
VI. $25(20-h)^{2} \Delta h$
A hemisphere of radius $r=4$

Donald Yeh

Numerade Educator

Problem 50

Explain what is wrong with the statement.
To find the area between the line $y=2 x,$ the $y$ -axis, and the line $y=8$ using horizontal slices, evaluate the integral $\int_{0}^{8} 2 y d y$

Gregory Higby

Numerade Educator

Problem 51

Explain what is wrong with the statement.
The volume of the sphere of radius 10 centered at the origin is given by the integral $\int_{-10}^{10} \pi \sqrt{10^{2}-x^{2}} d x$

Gregory Higby

Numerade Educator

Problem 52

Give an example of:
A region in the plane where it is easier to compute the area using horizontal slices than it is with vertical slices. Sketch the region.

Gregory Higby

Numerade Educator

Problem 53

Explain what is wrong with the statement.
A triangular region in the plane for which both horizontal and vertical slices work just as easily.

Gregory Higby

Numerade Educator

Problem 54

Are the statements true or false? Give an explanation for your answer.
The integral $\int_{-3}^{3} \pi\left(9-x^{2}\right) d x$ represents the volume of a sphere of radius 3 .

Gregory Higby

Numerade Educator

Problem 55

Are the statements true or false? Give an explanation for your answer.
The integral $\int_{0}^{h} \pi(r-y) d y$ gives the volume of a cone of radius $r$ and height $h$

Gregory Higby

Numerade Educator

Problem 56

Are the statements true or false? Give an explanation for your answer.
The integral $\int_{0}^{r} \pi \sqrt{r^{2}-y^{2}} d y$ gives the volume of a hemisphere of radius $r .$

Gregory Higby

Numerade Educator

Problem 57

Are the statements true or false? Give an explanation for your answer.
A cylinder of radius $r$ and length $l$ is lying on its side. Horizontal slicing tells us that the volume is given by $\int_{-r}^{r} 2 l \sqrt{r^{2}-y^{2}} d y$

Gregory Higby

Numerade Educator

Problem 58

Are the statements true or false? Give an explanation for your answer.
A cone of height 10 has a horizontal base with radius $50 .$ If $h$ is the height of a horizontal slice of thickness $\Delta h,$ the slice's volume is approximated by $\pi(50-5 h)^{2} \Delta h$

Gregory Higby

Numerade Educator

Problem 59

Are the statements true or false? Give an explanation for your answer.
A semicircle of radius 10 has a horizontal base. If $h$ is the height of a horizontal strip of thickness $\Delta h,$ the strip's area is approximated by $2 \sqrt{100-h^{2}} \Delta h$

Gregory Higby

Numerade Educator