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Thomas Calculus 12

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

Chapter 5

Integration

Educators


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 cqual width.
d. an upper sum with four rectangles of equal width.
$f(x)=x^{2}$ between $x=0$ and $x=1$

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Problem 2

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

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Problem 3

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

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Problem 4

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

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Problem 5

Using rectangles 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

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

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Problem 7

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

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Problem 8

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

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

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

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

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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).
a. Use rectangles to estimate how far the car traveled during the 36 sec it took to reach 142 mi/h.
b. Roughly how many seconds did it take the car to reach the halfway point? About how fast was the car going then?

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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 ower time
because of air resistance. The acceleration is measured in $\mathrm{ft} / \mathrm{sec}^{2}$ and recorded every second after the drop for $5 \mathrm{sec},$ as shown:
$\frac{t}{a} \left| \begin{array}{ccccc}{0} & {1} & {2} & {3} & {4} & {5} \\ \hline a & {32.00} & {19.41} & {11.77} & {7.14} & {4.33} & {2.63}\end{array}\right.$
a. Find an upper estimate for the speed when $t=5$
b. Find a lower estimate for the spced when $t=5$ .
c. Find an upper estimate for the distance fallen when $t=3$ .

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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 $\mathrm{ft} / \mathrm{sec}$ .
a. Assuming that gravity is the only force acting on the object, give an upper estimate for its velocity after 5 sec have elapsed. Use $g=32 \mathrm{ft} / \mathrm{sec}^{2}$ for the gravitational acceleration.
b. Find a lower estimate for the height attaincd after 5 $\mathrm{sec.}$

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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.
$f(x)=x^{3}$ on $[0,2]$

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Problem 16

$$f(x)=1 / x \quad \text { on } \quad[1,9]$$

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Problem 17

$$f(t)=(1 / 2)+\sin ^{2} \pi t \text { on }[0,2]$$

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Problem 18

$$f(t)=1-\left(\cos \frac{\pi t}{4}\right)^{4} \text { on }[0,4]$$

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Problem 19

$$\begin{array}{l}{\text { Water pollution Oil is leaking out of a tanker damaged at sea. }} \\ {\text { The damage to the tanker is worsening as evidenced by the in- }} \\ {\text { creased leakage each hour, recorded in the following table. }}\end{array}$$
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 cscaped after 8 hours.
c. The tanker continucs to leak 720 gal/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?

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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 relcased 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 Junc. 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?

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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) $\quad$ b. 8 (octagon) $\quad$ c. 16
d. Compare the areas in parts $(\mathrm{a}),$ (b), and (c) with the area of the circle.

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Problem 22

(Continuation 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 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 $r .$

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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.
d. Solve the equation $f(x)=($ average value) for $x$ using the average value calculated in part (c) for the $n=1000$ partitioning.
$f(x)=\sin x \quad$ on $\quad[0, \pi]$

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Problem 24

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

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Problem 25

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

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Problem 26

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

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