1 + (1/2)2 1 + (3/4)2 1 + 12

1

≈ · {1 + 3.7647 + 1.6 + 2.56 + 0.5}

12

≈ 0.785392.

Comparing with Example 8.30, we see that this answer is accurate to four

decimal places. We invite the reader to do the necessary calculation with the

Simpson™s Rule error to term to con¬rm that we could have predicted this

degree of accuracy.

You Try It: Estimate the integral

e2 1

dx

ln x

e

using both the Trapezoid Rule and Simpson™s Rule with a partition having six

points. Use the error term estimate to state what the accuracy prediction of each of

your calculations is. If the software Mathematica or Maple is available to you,

check the answers you have obtained against those provided by these computer

algebra systems.

260 CHAPTER 8 Applications of the Integral

Exercises

1. A solid has base the unit circle and vertical slices, parallel to the y-axis,

which are half-disks. Calculate the volume of this solid.

2. A solid has base a unit square with center at the origin and vertices on the

x- and y-axes. The vertical cross-section of this solid, parallel to the y-axis,

is a disk. What is the volume of this solid?

3. Set up the integral to calculate the volume enclosed when the indicated

curve over the indicated interval is rotated about the indicated line. Do not

evaluate the integral.

y = x 2 2 ¤ x ¤ 5 x-axis

(a)

√

y = x 1 ¤ x ¤ 9 y-axis

(b)

y = x 3/2 0 ¤ x ¤ 2 y = ’1

(c)

y = x + 3 ’1 ¤ x ¤ 2 y = 5

(d)

y = x 1/2 4 ¤ x ¤ 6 x = ’2

(e)

y = sin x 0 ¤ x ¤ π/2 y = 0

(f )

4. Set up the integral to evaluate the indicated surface area. Do not evaluate.

The area of the surface obtained when y = x 2/3 , 0 ¤ x ¤ 4, is

(a)

rotated about the x-axis.

The area of the surface obtained when y = x 1/2 , 0 ¤ x ¤ 3, is

(b)

rotated about the y-axis.

The area of the surface obtained when y = x 2 , 0 ¤ x ¤ 3, is rotated

(c)

about the line y = ’2.

The area of the surface obtained when y = sin x, 0 ¤ x ¤ π, is

(d)

rotated about the x-axis.

The area of the surface obtained when y = x 1/2 , 1 ¤ x ¤ 4, is

(e)

rotated about the line x = ’2.

The area of the surface obtained when y = x 3 , 0 ¤ x ¤ 1, is rotated

(f )

about the x-axis.

5. A water tank has a submerged window that is in the shape of a circle of

radius 2 feet. The center of this circular window is 8 feet below the surface.

Set up, but do not calculate, the integral for the pressure on the lower half

of this window”assuming that water weighs 62.4 pounds per cubic foot.

6. Aswimming pool is V-shaped. Each end of the pool is an inverted equilateral

triangle of side 10 feet. The pool is 25 feet long. The pool is full. Set up,

but do not calculate, the integral for the pressure on one end of the pool.

7. A man climbs a ladder with a 100 pound sack of sand that is leaking one

pound per minute. If he climbs steadily at the rate of 5 feet per minute, and

261

CHAPTER 8 Applications of the Integral

if the ladder is 40 feet high, then how much work does he do in climbing

the ladder?

8. Because of a prevailing wind, the force that opposes a certain runner is

3x 2 + 4x + 6 pounds at position x. How much work does this runner

perform as he runs from x = 3 to x = 100 (with distance measured in

feet)?

9. Set up, but do not evaluate, the integrals for each of the following arc length

problems.

The length of the curve y = sin x, 0 ¤ x ¤ π

(a)

The length of the curve x 2 = y 3 , 1 ¤ x ¤ 8

(b)

The length of the curve cos y = x, 0 ¤ y ¤ π/2

(c)

The length of the curve y = x 2 , 1 ¤ x ¤ 4

(d)

10. Set up the integral for, but do not calculate, the average value of the given

function on the given interval.

f (x) = sin2 x [2, 5]

(a)

g(x) = tan x [0, π/4]

(b)

x

(c) h(x) = , [’2, 2]

x+1

sin x

(d) f (x) = [’π, 2π ]

2 + cos x

11. Write down the sum that will estimate the given integral using the method

of rectangles with mesh of size k. You need not actually evaluate the sum.

4

e’x dx

2

k=6

(a)

0

2

k = 10

sin(ex ) dx

(b)

’2

0

k=5

cos x 2 dx

(c)

’2

4 ex

k = 12

(d) dx

2 + sin x

0

12. Do each of the problems in Exercise 11 with “method of rectangles” replaced

by “trapezoid rule.”

13. Do each of the problems in Exercise 11 with “method of rectangles” replaced

by “Simpson™s Rule.”

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BIBLIOGRAPHY

[CRC] Zwillinger et al., CRC Press Handbook of Tables and Formulas,

34th ed., CRC Press, Boca Raton, Florida, 1997.

[SCH1] Robert E. Moyer and Frank Ayres, Jr., Schaum™s Outline of

Trigonometry, McGraw-Hill, New York, 1999.

[SCH2] Fred Sa¬er, Schaum™s Outline of Precalculus, McGraw-Hill,

New York, 1997.

[SAH] S. L. Salas and E. Hille, Calculus, John Wiley and Sons, New York,

1982.

263

Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

This page intentionally left blank.

SOLUTIONS

TO EXERCISES

This book has a great many exercises. For some we provide

sketches of solutions and for others we provide just the answers.

For some, where there is repetition, we provide no answer. For the

sake of mastery, we encourage the student to write out complete

solutions to all problems.

Chapter 1

’5

1. (a)

24

43219445

(b)

1000000

’148

(c)

3198

19800

(d)

34251

’73162442

(e)

999000

’108

(f)

705

265

Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

266 Solutions to Exercises

14

(g)

885

32115422

(h)

9990000

√ √

2. In Fig. S1.2, set A = 3.4, B = ’π/2, C = 2π , D = ’ 2+1, E = 3·4,

F = 9/2, G = ’29/10.

G B D A F CE

x

0

Fig. S1.2

3.

Fig. S1.3(a)

Fig. S1.3(b)

Fig. S1.3(c)

Fig. S1.3(d)

Fig. S1.3(e)

Fig. S1.3(f)

267

Chapter 1

√√ √

4. Let A = (2, ’4), B = (’6, 3), C = (π, π 2 ), D = (’ 5, 8), E = ( 2π, ’3),

F = (1/3, ’19/4).

y

C

B D

x

E

A

F

Fig. S1.4

5.

Fig. S1.5(a) Fig. S1.5(b)

268 Solutions to Exercises

Fig. S1.5(d)

Y

Fig. S1.5(c)

FL

AM

TE

Fig. S1.5(f)

Fig. S1.5(e)

269

Chapter 1

6.

Fig. S1.6(a) Fig. S1.6(b)

Fig. S1.6(c)

270 Solutions to Exercises

Fig. S1.6(d)

Fig. S1.6(f)

Fig. S1.6(e)

271

Chapter 1

4’6 ’2

slope = =

7. (a)

2 ’ (’5) 7

4’2

= 1 hence requested line has slope ’1

(b) Given line has slope

3’1

Write y = ’(3/2)x + 3 hence slope is ’3/2

(c)

Write x ’ 4y = 6x + 6y or y = (’1/2)x hence slope is ’1/2

(d)

9’1 ’8

slope = =

(e)

(’8) ’ 1 9

Write y = x ’ 4 hence slope is 1

(f)

Slope is ’3/8 hence line is y ’ (’9) = (’3/8) · (x ’ 4)

8. (a)

Slope is 1 hence line is y ’ (’8) = 1 · (x ’ (’4))

(b)

y ’ 6 = (’8)(x ’ 4)

(c)

3’4 1

= ’ hence line is y ’ 3 = (’1/8)(x ’ 2)

(d) Slope is

2 ’ (’6) 8

y = 6x

(e)

Slope is ’3 hence line is y ’ 7 = (’3)(x ’ (’4))

(f)

9.

y y

x

x

(a) (b)

y

y

x x

(c) (d)

272 Solutions to Exercises

y

y

x x

(e) (f)

10. (a) Each person has one and only one father. This is a function.

(b) Some men have more than one dog, others have none. This is not a

function.

(c) Some real numbers have two square roots while others have none.