Calculus

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

Chapter 5

Key Concept: The Definite Integra - all with Video Answers

Educators

Section 1

How Do We Measure Distance Traveled?

Problem 1

Figure 5.11 shows the velocity of a car for $0 \leq t \leq 12$ and the rectangles used to estimate the distance traveled.
(a) Do the rectangles represent a left or a right sum?
(b) Do the rectangles lead to an upper or a lower estimate?
(c) What is the value of $n ?$
(d) What is the value of $\Delta t ?$
(e) Give an approximate value for the estimate.
(FIGURE CAN'T COPY)

Pawan Yadav

Pawan Yadav

Numerade Educator

Problem 2

Figure 5.12 shows the velocity of a car for $0 \leq t \leq 24$ and the rectangles used to estimate the distance traveled.
(a) Do the rectangles represent a left or a right sum?
(b) Do the rectangles lead to an upper or a lower estimate?
(c) What is the value of $n ?$
(d) What is the value of $\Delta t ?$
(e) Estimate the distance traveled.
(FIGURE CAN'T COPY)

Aman Gupta

Numerade Educator

Problem 3

Figure 5.13 shows the velocity of an object for $0 \leq t \leq$
6. Calculate the following estimates of the distance the object travels between $t=0$ and $t=6,$ and indicate whether each result is an upper or lower estimate of the distance traveled.
(a) A left sum with $n=2$ subdivisions
(b) A right sum with $n=2$ subdivisions
(FIGURE CAN'T COPY)

Sheryl Ezze

Numerade Educator

Problem 4

Figure 5.14 shows the velocity of an object for $0 \leq t \leq$
8. Calculate the following estimates of the distance the object travels between $t=0$ and $t=8,$ and indicate whether each is an upper or lower estimate of the distance traveled.
(a) A left sum with $n=2$ subdivisions
(b) A right sum with $n=2$ subdivisions
(FIGURE CAN'T COPY)

Sheryl Ezze

Numerade Educator

Problem 5

Figure 5.15 shows the velocity, $v,$ of an object (in meters/sec). Estimate the total distance the object traveled between $t=0$ and $t=6$
(FIGURE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 6

A bicyclist traveling at $20 \mathrm{ft} / \mathrm{sec}$ puts on the brakes to slow down at a constant rate, coming to a stop in 3 seconds.
(a) Figure 5.16 shows the velocity of the bike during braking. What are the values of $a$ and $b$ in the figure?
(b) How far does the bike travel while braking?
A bicyclist traveling at $20 \mathrm{ft} / \mathrm{sec}$ puts on the brakes to slow down at a constant rate, coming to a stop in 3 seconds.
(a) Figure 5.16 shows the velocity of the bike during braking. What are the values of $a$ and $b$ in the figure?
(b) How far does the bike travel while braking?
(FIGURE CAN'T COPY)

Aman Gupta

Numerade Educator

Problem 7

A bicyclist accelerates at a constant rate, from $0 \mathrm{ft} / \mathrm{sec}$
to $15 \mathrm{ft} / \mathrm{sec}$ in 10 seconds.
(a) Figure 5.17 shows the velocity of the bike while it
is accelerating. What is the value of $b$ in the figure?
(b) How far does the bike travel while it is accelerating?
(FIGURE CAN'T COPY)

Aman Gupta

Numerade Educator

Problem 8

A car accelerates at a constant rate from 44 ft/sec to 88 ft/sec in 5 seconds.
(a) Figure 5.18 shows the velocity of the car while it is accelerating. What are the values of $a, b$ and $c$ in the figure?
(b) How far does the car travel while it is accelerating?
(FIGURE CAN'T COPY)

Lucas Finney

Numerade Educator

Problem 9

A car slows down at a constant rate from $90 \mathrm{ft} / \mathrm{sec}$ to 20 ft/sec in 12 seconds.
(a) Figure 5.19 shows the velocity of the car while it is slowing down. What are the values of $a, b$ and $c$ in the figure?
(b) How far does the car travel while it is slowing down?
(FIGURE CAN'T COPY)

Aman Gupta

Numerade Educator

Problem 10

The velocity $v(t)$ in Table 5.3 is increasing, $0 \leq t \leq 12$
(a) Find an upper estimate for the total distance traveled using
(i) $\quad n=4$
(ii) $\quad n=2$
(b) Which of the two answers in part (a) is more accurate? Why?
(c) Find a lower estimate of the total distance traveled using $n=4$
(TABLE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 11

The velocity $v(t)$ in Table 5.4 is decreasing, $2 \leq t \leq 12$ Using $n=5$ subdivisions to approximate the total distance traveled, find
(a) An upper estimate
(b) A lower estimate
(TABLE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 12

A car comes to a stop five seconds after the driver applies the brakes. While the brakes are on, the velocities in the table are recorded.
(a) Give lower and upper estimates of the distance the car traveled after the brakes were applied.
(b) On a sketch of velocity against time, show the lower and upper estimates of part (a).
(c) Find the difference between the estimates. Explain how this difference can be visualized on the graph
in part (b).
(TABLE CAN'T COPY)

Aman Gupta

Numerade Educator

Problem 13

Table 5.5 gives the ground speed of a small plane accelerating for takeoff. Find upper and lower estimates for the distance traveled by the plane during takeoff.
(TABLE CAN'T COPY)

Aman Gupta

Numerade Educator

Problem 14

Figure 5.20 shows the velocity of a particle, in $\mathrm{cm} / \mathrm{sec}$, along a number line for time $-3 \leq t \leq 3$
(a) Describe the motion in words: Is the particle changing direction or always moving in the same direction? Is the particle speeding up or slowing down?
(b) Make over and underestimates of the distance traveled for $-3 \leq t \leq 3$
(TABLE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 15

At time, $t,$ in seconds, your velocity, $v,$ in meters/second, is given by
$$
v(t)=1+t^{2} \quad \text { for } \quad 0 \leq t \leq 6
$$
Use $\Delta t=2$ to estimate the distance traveled during this time. Find the upper and lower estimates, and then average the two.

Aman Gupta

Numerade Educator

Problem 16

For time, $t,$ in hours, $0 \leq t \leq 1,$ a bug is crawling at a velocity, $v,$ in meters/hour given by
$$
v=\frac{1}{1+t}
$$
Use $\Delta t=0.2$ to estimate the distance that the bug crawls during this hour. Find an overestimate and an underestimate. Then average the two to get a new estimate.

Aman Gupta

Numerade Educator

Problem 17

show the velocity, in $\mathrm{cm} / \mathrm{sec}$, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times $t=0$ and $t=5$ seconds.
(FIGURE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 18

show the velocity, in $\mathrm{cm} / \mathrm{sec}$, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times $t=0$ and $t=5$ seconds.
(FIGURE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 19

show the velocity, in $\mathrm{cm} / \mathrm{sec}$, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times $t=0$ and $t=5$ seconds.
(FIGURE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 20

show the velocity, in $\mathrm{cm} / \mathrm{sec}$, of a particle moving along a number line. (Positive velocities represent movement to the right; negative velocities to the left.) Find the change in position and total distance traveled between times $t=0$ and $t=5$ seconds.
(FIGURE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 21

The velocity of a car, in $\mathrm{ft} / \mathrm{sec},$ is $v(t)=10 t$ for $t$ in seconds, $0 \leq t \leq 6$
(a) Use $\Delta t=2$ to give upper and lower estimates for the distance traveled. What is their average?
(b) Find the distance traveled using the area under the graph of $v(t) .$ Compare it to your answer for part (a).

Aman Gupta

Numerade Educator

Problem 22

A particle moves with velocity $v(t)=3-t$ along the $x$ -axis, with time $t$ in seconds, $0 \leq t \leq 4$
(a) Use $\Delta t=1$ to give upper and lower estimates for the total displacement. What is their average?
(b) Graph $v(t) .$ Give the total displacement.

Aman Gupta

Numerade Educator

Problem 23

Use the expressions for left and right sums on page 277 and Table 5.6
(a) If $n=4,$ what is $\Delta t ?$ What are $t_{0}, t_{1}, t_{2}, t_{3}, t_{4} ?$ What are $f\left(t_{0}\right), f\left(t_{1}\right), f\left(t_{2}\right), f\left(t_{3}\right), f\left(t_{4}\right) ?$
(b) Find the left and right sums using $n=4$
(c) If $n=2,$ what is $\Delta t ?$ What are $t_{0}, t_{1}, t_{2} ?$ What are $f\left(t_{0}\right), f\left(t_{1}\right), f\left(t_{2}\right) ?$
(d) Find the left and right sums using $n=2$
(TABLE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 24

Use the expressions for left and right sums on page 277 and Table 5.7
(a) If $n=4,$ what is $\Delta t ?$ What are $t_{0}, t_{1}, t_{2}, t_{3}, t_{4} ?$ What are $f\left(t_{0}\right), f\left(t_{1}\right), f\left(t_{2}\right), f\left(t_{3}\right), f\left(t_{4}\right) ?$
(b) Find the left and right sums using $n=4$
(c) If $n=2,$ what is $\Delta t ?$ What are $t_{0}, t_{1}, t_{2} ?$ What are $f\left(t_{0}\right), f\left(t_{1}\right), f\left(t_{2}\right) ?$
(d) Find the left and right sums using $n=2$
(TABLE CAN'T COPY)

Noah Fisher

Numerade Educator

Problem 25

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff's data follow:
(TABLE CAN'T COPY)
(a) Assuming that Roger's speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour.
(b) Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half.
(c) How often would Jeff have needed to measure Roger's speed in order to find lower and upper estimates within 0.1 mile of the actual distance he ran?

Carson Merrill

Numerade Educator

Problem 26

The velocity of a particle moving along the $x$ -axis is given by $f(t)=6-2 t$ cm/sec. Use a graph of $f(t)$ to find the exact change in position of the particle from time $t=0$ to $t=4$ seconds.

Carson Merrill

Numerade Educator

Problem 27

find the difference between the upper and lower estimates of the distance traveled at velocity $f(t)$ on the interval $a \leq t \leq b$ for $n$ subdivisions.
$$f(t)=5 t+8, a=1, b=3, n=100$$

Carson Merrill

Numerade Educator

Problem 28

find the difference between the upper and lower estimates of the distance traveled at velocity $f(t)$ on the interval $a \leq t \leq b$ for $n$ subdivisions.
$$f(t)=25-t^{2}, a=1, b=4, n=500$$

Carson Merrill

Numerade Educator

Problem 29

find the difference between the upper and lower estimates of the distance traveled at velocity $f(t)$ on the interval $a \leq t \leq b$ for $n$ subdivisions.
$$f(t)=\sin t, a=0, b=\pi / 2, n=100$$

Carson Merrill

Numerade Educator

Problem 30

find the difference between the upper and lower estimates of the distance traveled at velocity $f(t)$ on the interval $a \leq t \leq b$ for $n$ subdivisions.
$$f(t)=e^{-t^{2} / 2}, a=0, b=2, n=20$$

Carson Merrill

Numerade Educator

Problem 31

A 2015 Porsche 918 Spyder accelerates from 0 to 88 ft/sec $(60 \mathrm{mph})$ in 2.2 seconds, the fastest acceleration of any car available for retail sale in $2015 .^{1}$
(a) Assuming that the acceleration is constant, graph the velocity from $t=0$ to $t=2.2$
(b) How far does the car travel during this time?

Carson Merrill

Numerade Educator

Problem 32

A baseball thrown directly upward at 96 ft/sec has velocity $v(t)=96-32 t$ ft/sec at time $t$ seconds.
(a) Graph the velocity from $t=0$ to $t=6$
(b) When does the baseball reach the peak of its flight? How high does it go?
(c) How high is the baseball at time $t=5 ?$

Carson Merrill

Numerade Educator

Problem 33

Figure 5.21 gives your velocity during a trip starting from home. Positive velocities take you away from home and negative velocities take you toward home. Where are you at the end of the 5 hours? When are you farthest from home? How far away are you at that time?
(FIGURE CAN'T COPY)

Nick Johnson

Numerade Educator

Problem 34

When an aircraft attempts to climb as rapidly as possible, its climb rate decreases with altitude. (This occurs because the air is less dense at higher altitudes.) The table shows performance data for a single-engine aircraft.
(TABLE CAN'T COPY)
(a) Calculate upper and lower estimates for the time required for this aircraft to climb from sea level to $10,000 \mathrm{ft}$
(b) If climb rate data were available in increments of $500 \mathrm{ft},$ what would be the difference between a lower and upper estimate of climb time based on
20 subdivisions?

Madi Sousa

Numerade Educator

Problem 35

Table 5.8 shows the upward vertical velocity $v(t),$ in ft/min, of a small plane at time $t$ seconds during a short flight.
(a) When is the plane going up? Going down?
(b) If the airport is located $110 \mathrm{ft}$ above sea level, estimate the maximum altitude the plane reaches during the flight.
(TABLE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 36

A bicyclist is pedaling along a straight road for one hour with a velocity $v$ shown in Figure $5.22 .$ She starts out five kilometers from the lake and positive velocities take her toward the lake. INote: The vertical lines on the graph are at 10 -minute ( $1 / 6$ -hour) intervals.]
(a) Does the cyclist ever turn around? If so, at what time(s)?
(b) When is she going the fastest? How fast is she going then? Toward the lake or away?
(c) When is she closest to the lake? Approximately how close to the lake does she get?
(d) When is she farthest from the lake? Approximately how far from the lake is she then?
(e) What is the total distance she traveled?
(GRAPH CAN'T COPY)

Adam Dehollander

Adam Dehollander

Numerade Educator

Problem 37

Two cars travel in the same direction along a straight road. Figure 5.23 shows the velocity, $v,$ of each car at time $t .$ Car $B$ starts 2 hours after car $A$ and car $B$ reaches a maximum velocity of $50 \mathrm{km} / \mathrm{hr}$
(a) For approximately how long does each car travel?
(b) Estimate car $A$ 's maximum velocity.
(c) Approximately how far does each car travel?
(GRAPH CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 38

Two cars start at the same time and travel in the same direction along a straight road. Figure 5.24 gives the velocity, $v,$ of each car as a function of time, $t .$ Which car:
(a) Attains the larger maximum velocity?
(b) Stops first?
(c) Travels farther?
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 39

A car initially going $50 \mathrm{ft} / \mathrm{sec}$ brakes at a constant rate (constant negative acceleration), coming to a stop in 5 seconds.
(a) Graph the velocity from $t=0$ to $t=5$
(b) How far does the car travel?
(c) How far does the car travel if its initial velocity is doubled, but it brakes at the same constant rate?

Khushbu Rani

Numerade Educator

Problem 40

A car moving with velocity $v$ has a stopping distance proportional to $v^{2}$
(a) If a car going 20 mi/hr has a stopping distance of 50 feet, what is its stopping distance going $40 \mathrm{mi} / \mathrm{hr}$ ? What about 60 milhr?
(b) After applying the brakes, a car going $30 \mathrm{ft} / \mathrm{sec}$ stops in 5 seconds and has $v=30-6 t .$ Explain why the stopping distance is given by the area under the graph of $v$ against $t$
(c) By looking at areas under graphs of $v$, explain why a car with the same deceleration as the car in part
(b) but an initial speed of $60 \mathrm{ft} / \mathrm{sec}$ has a stopping distance 4 times as far.

Carson Merrill

Numerade Educator

Problem 41

A woman drives 10 miles, accelerating uniformly from rest to 60 mph. Graph her velocity versus time. How long does it take for her to reach 60 mph?

Carson Merrill

Numerade Educator

Problem 42

An object has zero initial velocity and a constant acceleration of $32 \mathrm{ft} / \mathrm{sec}^{2} .$ Find a formula for its velocity as
a function of time. Use left and right sums with $\Delta t=1$ to find upper and lower bounds on the distance that the object travels in four seconds. Find the precise distance using the area under the curve.

Carson Merrill

Numerade Educator

Problem 43

explain what is wrong with the statement.
If a car accelerates from 0 to $50 \mathrm{ft} / \mathrm{sec}$ in 10 seconds, then it travels $250 \mathrm{ft}$

Carson Merrill

Numerade Educator

Problem 44

explain what is wrong with the statement.
For any acceleration, you can estimate the total distance traveled by a car in 1 second to within 0.1 feet by recording its velocity every 0.1 second.

Carson Merrill

Numerade Educator

Problem 45

give an example of:
A velocity function $f$ and an interval $[a, b]$ such that the distance denoted by the right-hand sum for $f$ on $[a, b]$ is less than the distance denoted by the left-hand sum, no matter what the number of subdivisions.

Carson Merrill

Numerade Educator

Problem 46

give an example of:
A velocity $f(t)$ and an interval $[a, b]$ such that at least 100 subdivisions are needed in order for the difference between the upper and lower estimates to be less than or equal to 0.1

Carson Merrill

Numerade Educator

Problem 47

Are the statements true or false? Give an explanation for your answer.
For an increasing velocity function on a fixed time interval, the left-hand sum with a given number of subdivisions is always less than the corresponding right-hand
sum.

Carson Merrill

Numerade Educator

Problem 48

Are the statements true or false? Give an explanation for your answer.
For a decreasing velocity function on a fixed time interval, the difference between the left-hand sum and righthand sum is halved when the number of subdivisions is doubled.

Carson Merrill

Numerade Educator

Problem 49

Are the statements true or false? Give an explanation for your answer.
For a given velocity function on a given interval, the difference between the left-hand sum and right-hand sum gets smaller as the number of subdivisions gets larger.

Carson Merrill

Numerade Educator