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Thomas Calculus

George B. Thomas, Jr.

Chapter 5

Integrals

Educators


Problem 1

In Exercises $1-4,$ use finite approximations to estimate the area under
the graph of the function using
$$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$
$f(x)=x^{2}$ between $x=0$ and $x=1$

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Problem 2

In Exercises $1-4,$ use finite approximations to estimate the area under
the graph of the function using
$$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$
$f(x)=x^{3}$ between $x=0$ and $x=1$

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Problem 3

In Exercises $1-4,$ use finite approximations to estimate the area under
the graph of the function using
$$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$
$f(x)=1 / x$ between $x=1$ and $x=5$

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Problem 4

In Exercises $1-4,$ use finite approximations to estimate the area under
the graph of the function using
$$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$
$f(x)=4-x^{2}$ between $x=-2$ and $x=2$

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Problem 5

Using rectangles each of whose height is given by the value of
the function at the midpoint of the rectangle's base (the midpoint
rule), estimate the area under the graphs of the following functions,
using first two and then four rectangles.
$f(x)=x^{2}$ between $x=0$ and $x=1$

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Problem 6

Using rectangles each of whose height is given by the value of
the function at the midpoint of the rectangle's base (the midpoint
rule), estimate the area under the graphs of the following functions,
using first two and then four rectangles.
$f(x)=x^{3}$ between $x=0$ and $x=1$

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Problem 7

Using rectangles each of whose height is given by the value of
the function at the midpoint of the rectangle's base (the midpoint
rule), estimate the area under the graphs of the following functions,
using first two and then four rectangles.
$f(x)=1 / x$ between $x=1$ and $x=5$

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Problem 8

Using rectangles each of whose height is given by the value of
the function at the midpoint of the rectangle's base (the midpoint
rule), estimate the area under the graphs of the following functions,
using first two and then four rectangles.
$f(x)=4-x^{2}$ between $x=-2$ and $x=2$

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Problem 9

Distance traveled The accompanying table shows the velocity
of a model train engine moving along a track for 10 sec. Estimate
the distance traveled by the engine using 10 subintervals of length
1 with
$$
\begin{array}{l}{\text { a. left-endpoint values. }} \\ {\text { b. right-endpoint values. }}\end{array}
$$

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Problem 10

Distance traveled upstream You are sitting on the bank of a
tidal river watching the incoming tide carry a bottle upstream. You
record the velocity of the flow every 5 minutes for an hour, with the
results shown in the accompanying table. About how far upstream did the bottle travel during that hour? Find an estimate using
12 subintervals of length 5 with
$$
\begin{array}{l}{\text { a. left-endpoint values. }} \\ {\text { b. right-endpoint values. }}\end{array}
$$

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Problem 11

Length of a road You and a companion are about to drive a
twisty stretch of dirt road in a car whose speedometer works but
whose odometer (mileage counter) is broken. To find out how
long this particular stretch of road is, you record the car's velocity at 10 -sec intervals, with the results shown in the accompanying
table. Estimate the length of the road using
$$
\begin{array}{l}{\text { a. left-endpoint values. }} \\ {\text { b. right-endpoint values. }}\end{array}
$$

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Problem 12

Distance from velocity data The accompanying table gives
data for the velocity of a vintage sports car accelerating from 0 to
142 $\mathrm{mi} / \mathrm{h}$ in 36 $\mathrm{sec}(10$ thousandths of an hour).
$$
\begin{array}{l}{\text { a. Use rectangles to estimate how far the car traveled during the }} \\ {36 \text { sec it took to reach } 142 \mathrm{mi} / \mathrm{h} \text { . }} \\ {\text { b. Roughly how many seconds did it take the car to reach the }} \\ {\text { halfway point? About how fast was the car going then? }}\end{array}
$$

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Problem 13

Free fall with air resistance An object is dropped straight down
from a helicopter. The object falls faster and faster but its acceleration (rate of change of its velocity) decreases over time because of
air resistance. The acceleration is measured in $\mathrm{ft} / \sec ^{2}$ and
recorded every second after the drop for $5 \mathrm{sec},$ as shown:
$$
\begin{array}{l}{\text { a. Find an upper estimate for the speed when } t=5 \text { . }} \\ {\text { b. Find a lower estimate for the speed when } t=5 \text { . }} \\ {\text { c. Find an upper estimate for the distance fallen when } t=3 \text { . }}\end{array}
$$

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Problem 14

Distance traveled by a projectile An object is shot straight
upward from sea level with an initial velocity of 400 $\mathrm{ft} / \mathrm{sec}$ .
$$
\begin{array}{l}{\text { a. Find an upper estimate for the speed when } t=5 \text { . }} \\ {\text { b. Find a lower estimate for the speed when } t=5 \text { . }} \\ {\text { c. Find an upper estimate for the distance fallen when } t=3 \text { . }}\end{array}
$$

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Problem 15

In Exercises 15-18, use a finite sum to estimate the average value of $f$
on the given interval by partitioning the interval into four subintervals
of equal length and evaluating $f$ at the subinterval midpoints.
$$
f(x)=x^{3} \quad \text { on } \quad[0,2]
$$

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Problem 16

In Exercises 15-18, use a finite sum to estimate the average value of $f$
on the given interval by partitioning the interval into four subintervals
of equal length and evaluating $f$ at the subinterval midpoints.
$$
f(x)=1 / x \text { on }[1,9]
$$

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Problem 17

In Exercises 15-18, use a finite sum to estimate the average value of $f$
on the given interval by partitioning the interval into four subintervals
of equal length and evaluating $f$ at the subinterval midpoints.
$$
f(t)=(1 / 2)+\sin ^{2} \pi t \quad \text { on } \quad[0,2]
$$

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Problem 18

In Exercises 15-18, use a finite sum to estimate the average value of $f$
on the given interval by partitioning the interval into four subintervals
of equal length and evaluating $f$ at the subinterval midpoints.
$$
f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \quad \text { on } \quad[0,4]
$$

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Problem 19

Water pollution Oil is leaking out of a tanker damaged at sea. The
damage to the tanker is worsening as evidenced by the increased
leakage each hour, recorded in the following table.
a. Give an upper and a lower estimate of the total quantity of oil
that has escaped after 5 hours.
b. Repeat part (a) for the quantity of oil that has escaped after
8 hours.
c. The tanker continues to leak 720 gal $/ \mathrm{h}$ after the first 8 hours.
If the tanker originally contained $25,000$ gal of oil, approximately how many more hours will elapse in the worst case
before all the oil has spilled? In the best case?

James M.
Numerade Educator

Problem 20

Air pollution A power plant generates electricity by burning oil.
Pollutants produced as a result of the burning process are removed
by scrubbers in the smokestacks. Over time, the scrubbers
become less efficient and eventually they must be replaced when the amount of pollution released exceeds government standards.
Measurements are taken at the end of each month determining the
rate at which pollutants are released into the atmosphere, recorded
as follows.
a. Assuming a $30-$ day month and that new scrubbers allow only
0.05 ton/day to be released, give an upper estimate of the
total tonnage of pollutants released by the end of June. What is
a lower estimate?
b. In the best case, approximately when will a total of 125 tons
of pollutants have been released into the atmosphere?

Danny L.
Numerade Educator

Problem 21

Inscribe a regular $n$ -sided polygon inside a circle of radius 1 and
compute the area of the polygon for the following values of $n :$
$$
\begin{array}{l}{\text { a. } 4 \text { (square) } \quad \text { b. } 8 \text { (octagon) } \quad \text { c. } 16} \\ {\text { d. Compare the areas in parts (a), (b), and (c) with the area of the }} \\ {\text { circle. }}\end{array}
$$

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Problem 22

(Contimuation of Exercise $21 . )$
a. Inscribe a regular $n$ -sided polygon inside a circle of radius 1 and
compute the area of one of the $n$ congruent trriangles formed by
drawing radii to the vertices of the polygon.
b. Compute the limit of the area of the inscribed polygon as
$n \rightarrow \infty$
c. Repeat the computations in parts (a) and (b) for a circle of
radius $r .$

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Problem 23

In Exercises $23-26,$ use a CAS to perform the following steps.
a. Plot the functions over the given interval.
b. Subdivide the interval into $n=100,200,$ and 1000 subintervals of equal length and evaluate the function at the midpoint
of each subinterval.
c. Compute the average value of the function values generated in
part (b).
d. Solve the equation $f(x)=($ average value) for $x$ using the average value calculated in part (c) for the $n=1000$ partitioning.
$$
f(x)=\sin x \quad \text { on } \quad[0, \pi]
$$

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Problem 24

In Exercises $23-26,$ use a CAS to perform the following steps.
a. Plot the functions over the given interval.
b. Subdivide the interval into $n=100,200,$ and 1000 subintervals of equal length and evaluate the function at the midpoint
of each subinterval.
c. Compute the average value of the function values generated in
part (b).
d. Solve the equation $f(x)=($ average value) for $x$ using the average value calculated in part (c) for the $n=1000$ partitioning.
$$
f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi]
$$

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Problem 25

In Exercises $23-26,$ use a CAS to perform the following steps.
a. Plot the functions over the given interval.
b. Subdivide the interval into $n=100,200,$ and 1000 subintervals of equal length and evaluate the function at the midpoint
of each subinterval.
c. Compute the average value of the function values generated in
part (b).
d. Solve the equation $f(x)=($ average value) for $x$ using the average value calculated in part (c) for the $n=1000$ partitioning.
$$
f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right] \quad 2
$$

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Problem 26

In Exercises $23-26,$ use a CAS to perform the following steps.
a. Plot the functions over the given interval.
b. Subdivide the interval into $n=100,200,$ and 1000 subintervals of equal length and evaluate the function at the midpoint
of each subinterval.
c. Compute the average value of the function values generated in
part (b).
d. Solve the equation $f(x)=($ average value) for $x$ using the average value calculated in part (c) for the $n=1000$ partitioning.
$$
f(x)=x \sin ^{2} \frac{1}{x} \quad \text { on } \quad\left\lfloor\frac{\pi}{4}, \pi\right]
$$

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