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Thomas' Calculus: Early Transcendentals in SI Units

George B. Thomas Jr., Maurice D. Weir, Joel R. Hass

Chapter 5

Integrals - all with Video Answers

Educators

Section 1

Area and Estimating with Finite Sums

10:18

Problem 1

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

Regina Hays
Regina Hays
Numerade Educator
10:18

Problem 2

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

Regina Hays
Regina Hays
Numerade Educator
03:10

Problem 3

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

Gregory Higby
Gregory Higby
Numerade Educator
10:18

Problem 4

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

Regina Hays
Regina Hays
Numerade Educator
04:44

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.
$$
f(x)=x^{2} \text { between } x=0 \text { and } x=1
$$

Chris Smith
Chris Smith
Numerade Educator
04:04

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.
$$
f(x)=x^{3} \text { between } x=0 \text { and } x=1
$$

Chris Smith
Chris Smith
Numerade Educator
04:25

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.
$$
f(x)=1 / x \text { between } x=1 \text { and } x=5
$$

Chris Smith
Chris Smith
Numerade Educator
04:25

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.
$$
f(x)=4-x^{2} \text { between } x=-2 \text { and } x=2
$$

Chris Smith
Chris Smith
Numerade Educator
04:18

Problem 9

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

Chris Smith
Chris Smith
Numerade Educator
05:49

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
a. left-endpoint values.
b. right-endpoint values.

Chris Smith
Chris Smith
Numerade Educator
05:12

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 (kilometer counter) is broken. To find out how long this particular stretch of road is, you record the car's velocity at 10 -second intervals, with the results shown in the accompanying table. Estimate the length of the road using
a. left-endpoint values.
b. right-endpoint values.

Regina Hays
Regina Hays
Numerade Educator
06:13

Problem 12

Distance from velocity data The accompanying table gives data for the velocity of a vintage sports car accelerating from 0 to $228 \mathrm{~km} / \mathrm{h}$ in $36 \mathrm{~s}$ ( 10 thousandths of an hour).
a. Use rectangles to estimate how far the car traveled during the $36 \mathrm{~s}$ it took to reach $228 \mathrm{~km} / \mathrm{h}$.
b. Roughly how many seconds did it take the car to reach the halfway point? About how fast was the car going then?

Brittany Knowlton
Brittany Knowlton
Numerade Educator
08:34

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 $\mathrm{m} / \mathrm{s}^{2}$ and recorded every second after the drop for $5 \mathrm{~s}$, as shown:
a. Find an upper estimate for the speed when $t=5$.
b. Find a lower estimate for the speed when $t=5$.
c. Find an upper estimate for the distance fallen when $t=3$.

Regina Hays
Regina Hays
Numerade Educator
03:23

Problem 14

Distance traveled by a projectile An object is shot straight upward from sea level with an initial velocity of $122.5 \mathrm{~m} / \mathrm{s}$.
a. Assuming that gravity is the only force acting on the object, give an upper estimate for its velocity after $5 \mathrm{~s}$ have elapsed. Use $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$ for the gravitational acceleration.
b. Find a lower estimate for the height attained after $5 \mathrm{~s}$.

Massimo Antonelli
Massimo Antonelli
Numerade Educator
07:02

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 sub intervals of equal length and evaluating $f$ at the sub interval midpoints.
$$
f(x)=x^{3} \quad \text { on } \quad[0,2]
$$

Nick Johnson
Nick Johnson
Numerade Educator
04:09

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 sub intervals of equal length and evaluating $f$ at the sub interval midpoints.
$$
f(x)=1 / x \text { on }[1,9]
$$

RC
Royce Christensen
Numerade Educator
02:38

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 sub intervals of equal length and evaluating $f$ at the sub interval midpoints.
$$
f(t)=(1 / 2)+\sin ^{2} \pi t \text { on }[0,2]
$$

PD
Philip Dzienciol
Numerade Educator
04:53

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 sub intervals of equal length and evaluating $f$ at the sub interval midpoints.
$$
f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \text { on }[0,4]
$$

Brittany Knowlton
Brittany Knowlton
Numerade Educator
07:02

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 $720 \mathrm{~L} / \mathrm{h}$ after the first 8 hours. If the tanker originally contained $25,000 \mathrm{~L}$ of oil, approximately how many more hours will elapse in the worst case before all the oil has spilled? In the best case?

Chris Smith
Chris Smith
Numerade Educator
05:09

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 tonne $/$ 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 tonnes of pollutants have been released into the atmosphere?

Charles Carter
Charles Carter
Numerade Educator
04:39

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 $n$ :
a. 4 (square)
b. 8 (octagon)
c. 16
d. Compare the areas in parts (a), (b), and (c) with the area of the circle.

Chris Smith
Chris Smith
Numerade Educator
06:13

Problem 22

a. Inscribe a regular $n$ -sided polygon inside a circle of radius 1 and compute the area of one of the $n$ congruent triangles 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 $r$.

Chris Smith
Chris Smith
Numerade Educator
09:14

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.
$$
f(x)=\sin x \quad \text { on } \quad[0, \pi]
$$

Regina Hays
Regina Hays
Numerade Educator
10:11

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.
$$
f(x)=\sin ^{2} x \quad \text { on } \quad[0, \pi]
$$

Regina Hays
Regina Hays
Numerade Educator
08:46

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.
$$
f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right] 2
$$

Regina Hays
Regina Hays
Numerade Educator
08:46

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.
$$
f(x)=x \sin ^{2} \frac{1}{x} \quad \text { on }\left[\frac{\pi}{4}, \pi\right]
$$

Regina Hays
Regina Hays
Numerade Educator