Name: Class: Date: ID: A Multiple Choice

65
Name: ________________________ Class: ___________________ Date: __________ ID: A 1 104ex1questions Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Find the value of the integral sin 2t ( ) dt. 0 π 2 a. 0 e. 2 4 b. 1 f. 2 c. 3 2 g. 2 2 d. π 2 h. 1 2 ____ 2. Find the value of the integral x 3 + 1 Ê Ë Á Á Á ˆ ¯ ˜ ˜ ˜ 5 x 2 dx. 0 1 a. 0 e. 32 3 b. 1 f. 2 c. 21 2 g. 7 2 d. 32 9 h. 3 2 ____ 3. Find the value of the integral x 2 x 3 + 1 Ê Ë Á Á Á ˆ ¯ ˜ ˜ ˜ 2 dx. 0 1 a. 3 4 e. 1 6 b. 2 f. 3 2 c. 3 7 g. 2 3 d. 7 3 h. 1

Transcript of Name: Class: Date: ID: A Multiple Choice

Name: ________________________ Class: ___________________ Date: __________ ID: A

1

104ex1questions

Multiple Choice

Identify the choice that best completes the statement or answers the question.

____ 1. Find the value of the integral sin 2t( ) dt.0

π 2

a. 0 e.2

4

b. 1 f. 2

c.3

2g.

2

2

d.π2

h.1

2

____ 2. Find the value of the integral x3

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

5

x2

dx.0

1

a. 0 e.32

3

b. 1 f. 2

c.21

2g.

7

2

d.32

9h.

3

2

____ 3. Find the value of the integral x

2

x3

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

2dx.

0

1

a.3

4e.

1

6

b. 2 f.3

2

c.3

7g.

2

3

d.7

3h. 1

Name: ________________________ ID: A

2

____ 4. Find the value of the integral sin2x cos x dx.

0

π 4

a.2

18e.

2

6

b.π4

f.2

9

c.3π2

g.2π3

d.2

12h.

π2

____ 5. Find the value of the integral sec x tanx 1 + sec x( ) dx.0

π 3

a. 4 e.9

2

b.5

2f. 2

c. 3 g.7

2

d.11

2h. 5

____ 6. Find the value of the integral 1

1 + xÊ

ËÁÁÁ

ˆ

¯˜̃˜

2

1

xdx.

1

4

a.6

5e.

4

9

b.1

3f.

3

2

c.2

3g.

5

6

d.5

2h.

1

6

____ 7. Find the value of the integral cos x sin sinx( ) dx.0

π 2

a.π2

e.π2

− sin1

b. 1 −π4

f.π4

+ cos 1

c. sin 1 g. 1 +3π4

d. 1 – cos 1 h. 1 + tan 1

Name: ________________________ ID: A

3

____ 8. Find the value of the integral x

1 + 3x2

dx.−3

3

a.2

37 − 1

Ê

ËÁÁÁ

ˆ

¯˜̃˜ e.

2

3

b. 7 − 1 f.1

7

c. 0 g. 7

d.1

3h. Does not exist

____ 9. Find the value of lnx( )

2

xdx.

e

e2

∫a. ln2 e. 1

b.1

2ln2 f. 1 ln2( )

c.1

2g. 0

d.3

2h.

7

3

____ 10. Find the value of dx

x lnx.

e

e4

a. 0 e. 4

b. 1 f. 5

c. 2 g. 6

d. 3 h. 7

____ 11. Find the value of the integral x2e

−x3

dx.0

1

a.−e

2e. −e

b.e

2f.

1 − e−1

3

c.1 − e

−1

2g.

e−1

2

d. e h. e−1

Name: ________________________ ID: A

4

____ 12. Find the value of the integral e

x

ex

+ 1dx.

0

1

a. e + 1 e.1

2ln e − 1( )

b. ln e − 1( ) f.e + 1

2

c.e − 1

2g.

1

2ln e + 1( )

d. lne + 1

2h. e − 1

____ 13. Find the value of e4x

dx.0

ln 3

a.45

4e.

41

4

b. 11 f. 10

c. 80 g. 20

d.21

2h.

19

2

____ 14. Find the value of the integral sin

−1x

1 − x2

dx.0

1 2

a.π 2

36e.

π 2

72

b.π36

f.π72

c.1

36g.

1

72

d.π6

h.π2

____ 15. Find the value of the integral 1

4 − 25x2

dx.0

2 5

a.π4

e.π20

b.π5

f.π40

c.π10

g.π2

d.1

4h.

1

20

Name: ________________________ ID: A

5

____ 16. Find the value of the integral 1

x2

− 8x + 17dx.

4 − 3

4 + 3

a.π4

e.π3

b.4π3

f.2π3

c.π10

g.π2

d.2

3h.

1

2

____ 17. Find the value of the integral xex

dx.0

1

∫a. 2 e. e

b. e2

− e f. e − 1

c. 1 g. e − 2

d. e2

h.e − 1

2

____ 18. Find the value of the integral lnx dx.1

e

∫a. e

2e. e − 2

b. e f. 1

c. 2 g. e − 1

d. e2

− e h.e − 1

2

____ 19. Find the value of the integral x cos x dx.0

π

∫a. −2 e. 2

b. 2π − 2 f. π

c.π2

g. 2π

d. 4 h. −4

Name: ________________________ ID: A

6

____ 20. Find the value of the integral ex

cos x dx.0

π 2

a.e

π 4+ 1

2e.

eπ 4

− 1

4

b.e

π 2+ 1

2f.

eπ 4

− 1

2

c.e

π 4+ 1

4g.

eπ 2

+ 1

4

d.e

π 2− 1

4h.

eπ 2

− 1

2

____ 21. Find the value of the integral lnx

x2

dx.1

4

∫a. e e. e − 2

b. 2e − 1 f. e − 1

c.3

2− ln2 g.

3

4−

ln2

2

d.1

2−

ln2

2h. 1 − ln2

____ 22. Find the value of the integral x tan−1

x dx.0

1

a.π4

e.π − 2

4

b. π − 2 f. π − 1

c.π2

g.π − 1

2

d.π − 2

2h.

π − 1

4

____ 23. Find the value of the integral x2

sinx dx.0

π 2

∫a. π − 2 e. 2 − π

b.π − 2

4f.

2 − π4

c.π − 2

2g.

2 − π2

d. 1 h. 0

Name: ________________________ ID: A

7

____ 24. Find the value of the integral arcsin t dt.0

1

∫a. π − 2 e. 2 − π

b.π − 2

4f.

2 − π4

c.π − 2

2g.

2 − π2

d. 1 h. 0

____ 25. Find the value of the integral t 2t − 1( )4

dt.0

1

a.1

3e.

1

2

b.1

4f.

1

10

c.1

5g.

1

20

d. 1 h. 0

____ 26. Find the value of the integral ln 1 + x2Ê

ËÁÁÁ

ˆ¯˜̃̃ dx.

0

1

a. ln2 e.π4

+ ln2

b.π8

f. π − 4

c.π2

− 2 + ln2 g. π − 2

d. 2 − ln2 h. π − ln2

____ 27. Evaluate the integral 1 + lnx

x lnxdx.∫

a. lnx + C e.x

lnx+ C

b. lnlnx + C f.lnx

x + lnx( )+ C

c. x + lnx + C g. x lnx + C

d. lnx + lnlnx + C h. x lnlnx + C

Name: ________________________ ID: A

8

____ 28. Evaluate the integral cos x dx.∫a. 2sin x + C e. 2 x sin x + cos x

Ê

ËÁÁÁ

ˆ

¯˜̃˜ + C

b. 2 x cos x + C f. 2 x cos x + sin xÊ

ËÁÁÁ

ˆ

¯˜̃˜ + C

c. x cos x + sin xÊ

ËÁÁÁ

ˆ

¯˜̃˜ + C g. x cos x +

sin x

x+ C

d.cos x + sin x

x+ C h. x sin x +

cos x

x+ C

____ 29. Evaluate cos lnx( )dx.∫a. sin lnx( ) + C e.

sinx

x+ C

b.x

2

2cos lnx( ) + sin lnx( )ÈÎÍÍÍ

˘˚˙̇˙ + C f. x cos lnx( ) + sin lnx( )

ÈÎÍÍÍ

˘˚˙̇˙ + C

c. cos1

x

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C g. −sin

1

x

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C

d.x

4cos lnx( ) + sin lnx( )ÈÎÍÍÍ

˘˚˙̇˙ + C h.

x

2cos lnx( ) + sin lnx( )ÈÎÍÍÍ

˘˚˙̇˙ + C

____ 30. Find the value of the integral 1 − u2

du.0

1

a. 2 e.1

2

b. 1 f. π

c.π4

g.π2

d.1

4h. 2π

____ 31. Find the value of the integral cos3x dx

0

π 2

∫a. 1 e. 3

b.1

3f.

π2

c.2

3g.

1

6

d. 2 h.π2

−π 3

24

Name: ________________________ ID: A

9

____ 32. Find the value of the integral sinx

cos3x

dx.0

π 6

a. 1 e.1

3

b. −2

3f. −

1

3

c.2

3g.

1

6

d. 2 h. −1

6

____ 33. Find the value of the integral u2

− 1 du.1

2

a. 3 +1

2ln 4 − 3Ê

ËÁÁÁ

ˆ

¯˜̃˜ e. 3 −

1

2ln 4 − 3Ê

ËÁÁÁ

ˆ

¯˜̃˜

b. 3 −1

2ln 2 + 3Ê

ËÁÁÁ

ˆ

¯˜̃˜ f. 3 +

1

2ln 4 + 3Ê

ËÁÁÁ

ˆ

¯˜̃˜

c. 3 −1

2ln 4 + 3Ê

ËÁÁÁ

ˆ

¯˜̃˜ g. 3 +

1

2ln 2 + 3Ê

ËÁÁÁ

ˆ

¯˜̃˜

d. 3 +1

2ln 2 − 3Ê

ËÁÁÁ

ˆ

¯˜̃˜ h. 3 −

1

2ln 2 − 3Ê

ËÁÁÁ

ˆ

¯˜̃˜

____ 34. Find the value of the integral sin2

4x( ) dx.0

π 8

a. 1 e.π32

b.π16

f.π8

c.π3

g.π6

d. 2 h.1

6

____ 35. Find the value of the integral sin2u du.

0

π 12

a.π − 3

18e.

π − 3

36

b.π − 3

16f.

π − 3

12

c.π − 3

24g.

π − 3

48

d.π − 3

72h.

π − 3

60

Name: ________________________ ID: A

10

____ 36. Find the value of the integral sin2

2x( ) cos2

2x( ) dx.0

π 4

a. 1 e.π32

b.π16

f.π8

c.π4

g.π64

d. 2 h.1

6

____ 37. Evaluate cos x

4 + sin2x

dx.∫

a. tan−1 sinx

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C e.

1

4tan

−1 sinx

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C

b.1

2tan

−1 sinx

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C f. ln

sinx

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C

c.1

2tan

−1sinx( ) + C g. 2tan

−1 sinx

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C

d.1

2tan

−1 cos x

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C h.

1

2ln

sinx

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C

____ 38. Evaluate 3x + 2

x − 1dx.∫

a. 5ln x − 1| | + 3x + C e. 3ln x − 1| | + 5x + C

b. 5ln x − 1| | + C f. 5x + C

c. 3ln x − 1| | + 3x + C g. 2ln x − 1| | + 3x + C

d. 3x + C h. 2ln x| | + C

____ 39. Find the partial fraction expansion of the rational function: x

x2

− 5x + 6.

a.3

x − 3−

2

x − 2e.

3

x + 3−

2

x + 2

b.3

x − 3+

2

x − 2f.

−3

x + 3−

2

x − 2

c.3

x + 3−

2

x − 2g.

−3

x − 3+

2

x − 2

d.3

x − 3−

2

x + 2h.

−3

x − 3+

−2

x − 2

Name: ________________________ ID: A

11

____ 40. Find the partial fraction expansion of the rational function: 9

x + 1( )2

2 − x( ).

a.−1

x + 1+

3

x + 1( )2

+−1

x − 2e.

1

x + 1+

3

x + 1( )2

+1

x − 2

b.1

x + 1−

1

x − 2f.

1

x + 1−

3

x + 1( )2

−1

x − 2

c.−1

x + 1+

−3

x + 1( )2

+−1

x − 2g.

1

x + 1+

3

x + 1( )2

−1

x − 2

d.3

x + 1( )2

−1

x − 2h.

−1

x + 1+

−3

x + 1( )2

+1

x − 2

____ 41. Find the partial fraction expansion of the rational function: x

2

x − 1( ) x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

.

a.1

2 x − 1( )+

1

2 x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

e.1

2 x − 1( )+

x + 1

2 x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

b.1

2 x − 1( )+

x

2 x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

f.−1

2 x − 1( )+

x + 1

2 x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

c.1

x − 1( )+

x + 1

x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

g.1

x − 1( )+

x

x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

d.1

x − 1( )+

1

x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

h.1

2 x − 1( )−

x + 1

2 x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

____ 42. Evaluate 1

x2

− 4dx.∫

a.1

4ln

x + 2

x − 2

|||

|||+ C e.

1

2ln

x + 2

x − 2

|||

|||+ C

b. ln x2

− 4||

|| + C f.

1

4ln x

2− 4

||

|| + C

c. 4lnx + 2

x − 2

|||

|||+ C g.

1

2ln

x − 2

x + 2

|||

|||+ C

d.1

4ln

x − 2

x + 2

|||

|||+ C h. −

1

x− 4x + C

Name: ________________________ ID: A

12

____ 43. Use the Trapezoidal Rule with n = 1 to approximate the integral x + 2Ê

ËÁÁÁ

ˆ

¯˜̃˜ dx.

0

1

a.1

2e.

3

8

b.9

16f.

2

3

c.7

16g.

5

2

d.1

4h.

5

8

____ 44. Use Simpson’s Rule with n = 2 to approximate the integral x3

dx.0

1

a.5

8e.

7

16

b.1

3f.

1

4

c.3

8g.

9

16

d.2

3h.

1

2

____ 45. Use the Trapezoidal Rule with n = 2 to approximate the integral x3

dx.0

1

a.5

16e.

1

3

b.1

4f.

7

16

c.1

2g.

2

3

d.5

8h.

3

8

____ 46. Use the Midpoint Rule with n = 2 to approximate the integral x3

dx.0

1

a.1

4e.

5

16

b.1

2f.

5

32

c.7

32g.

7

16

d.3

2h.

5

2

Name: ________________________ ID: A

13

____ 47. Use the Trapezoidal Rule with n = 4 to approximate the integral 1

xdx.

1

3

a.67

30e.

5

32

b.29

10f.

67

60

c.7

8g.

7

16

d.3

2h.

3

4

____ 48. Use the Midpoint Rule with n = 4 to approximate the integral 1

xdx.

1

3

a.3776

3465e.

5

16

b.7552

3465f.

5

32

c.1888

3465g.

7

16

d.7

32h.

5

2

____ 49. Use Simpson’s Rule with n = 4 to approximate the integral 1

xdx.

1

3

a.33

10e.

11

10

b.66

10f.

22

10

c.12

10g.

6

10

d.21

20h.

21

10

____ 50. Use Simpson’s Rule with n = 4 to approximate the integral 2x

dx.−2

2

a.65

4e.

65

12

b.1

2f.

45

4

c.45

8g.

7

16

d.3

2h.

5

2

Name: ________________________ ID: A

14

____ 51. Use the Trapezoidal Rule with n = 4 to approximate the integral 2x

dx.−2

2

a.65

4e.

65

12

b.1

2f.

45

4

c.45

8g.

7

16

d.3

2h.

5

2

____ 52. Use the Midpoint Rule with n = 4 to approximate the integral 2x

dx.−2

2

∫a. 16.25 e. 5.625

b. 0.505 f. 11.25

c. 5.4167 g. 5.3033

d. 1.5 h. 2.5

____ 53. Suppose using n = 10 to approximate the integral of a certain function by the Trapezoidal Rule results in an

upper bound for the error equal to 1

10. What will the upper bound become if we change to n = 20?

a.1

10,000e.

1

1000

b.1

100f.

1

20

c.1

80g.

1

40

d.1

160h.

1

100,000

____ 54. Suppose using n = 10 to approximate the integral of a certain function by Simpson’s Rule results in an upper

bound for the error equal to 1

10. What will the upper bound become if we change to n = 20?

a.1

100e.

1

160

b.1

10,000f.

1

20

c.1

40g.

1

100,000

d.1

1000h.

1

80

Name: ________________________ ID: A

15

____ 55. Which one of the following is not an improper integral?

a.1

x + 2dx

−4

2

∫ e.1

x − 2dx

−∞

1

b.1

x2

− 2dx

−2

2

∫ f.1

x + 2dx

−2

2

c.1

x + 2dx

0

∫ g.1

x2

+ 2dx

−∞

d.1

x2

+ 2dx

−2

2

∫ h.1

x − 2dx

−2

2

____ 56. Which one of the following integrals is improper?

a. 2 x dx0

2

∫ e. 3x

dx0

1

b.1

x2

− 2dx

−2

2

∫ f.1

x4

+ 1dx

−2

2

c.1

x + 2dx

0

3

∫ g.1

x2

+ 2dx

−5

5

d. lnx dx1

2

∫ h. None of the above

____ 57. Evaluate the improper integral x−2

1

∫ dx.

a.1

4e.

1

2

b. 1 f. 3

c. 2 g.1

3

d. 4 h. Divergent

____ 58. Evaluate the improper integral x−2

0

1

∫ dx.

a. 3 e.1

3

b.1

4f.

1

2

c. 4 g. 1

d. 2 h. Divergent

Name: ________________________ ID: A

16

____ 59. Evaluate the improper integral x− 1 2

0

∫ dx.

a. 4 e.1

2

b. 3 f. 2

c.1

4g. 1

d.1

3h. Divergent

____ 60. Evaluate the improper integral x−1 2

0

1

∫ dx.

a.1

3e.

1

4

b.1

2f. 3

c. 1 g. 4

d. 2 h. Divergent

____ 61. Evaluate the improper integral lnx

x1

∫ dx.

a. 2 e. ln2

b.1

2ln2 f.

1

2

c.1

4ln2 g. 2ln2

d. 1 h. Divergent

____ 62. Evaluate the improper integral lnx

x0

1

∫ dx.

a.1

2e.

1

2ln2

b. 1 f. 2ln2

c. ln2 g. 2

d.1

4ln2 h. Divergent

____ 63. Evaluate the improper integral xe−x

2

−∞

∫ dx.

a. e e. e−1

b. e2

− 1 f. e−2

c. 0 g. 1

d. e2

h. 1 − e−2

Name: ________________________ ID: A

17

____ 64. Evaluate the improper integral xe− x 2

0

∫ dx.

a. 1 e.1

3

b. 2 f. 3

c. 4 g.1

2

d.1

4h. Divergent

____ 65. Evaluate the improper integral e−3x

0

∫ dx.

a. 3 e. −1

b.1

3f. −3

c. −1

3g. 0

d. 1 h. Divergent

____ 66. Evaluate the improper integral xe−x

2

0

∫ dx.

a. 0 e. e − 1

b. 1 f.1

2

c. e g. 2

d. e−1

h. Divergent

____ 67. Evaluate the improper integral 1

4 − x21

2

∫ dx.

a.π3

e.π2

b. 0 f.π6

c. π g. 1

d.3

2h. Divergent

____ 68. Evaluate the improper integral 1

4 − x0

4

∫ dx.

a.π2

e.π16

b.π4

f.π8

c. π g. 1

d.3

2h. Divergent

Name: ________________________ ID: A

18

____ 69. Evaluate the improper integral sec2x

0

3π 4

∫ dx.

a. −1 e.π16

b.1

2f.

3

2

c. π g. 1

d. 2 h. Divergent

____ 70. Evaluate the improper integral cos x0

∫ dx.

a.π4

e.π16

b.π2

f.3

2

c. π g. 1

d. 2 h. Divergent

____ 71. Evaluate the improper integral 1

x x + 1( )1

∫ dx.

a.1

2ln4 e.

1

3ln4

b. ln4 f. ln2

c. −ln4 g. −1

2ln4

d. −1

3ln4 h. Divergent

____ 72. Find the area of the region bounded by the curves y = x2

+ 1 and y = 2

a.4

3e.

8

9

b.2

3f. 1

c.11

9g.

14

9

d.5

3h. 2

Name: ________________________ ID: A

19

____ 73. Find the area of the region bounded by the curves y = x2

− 4x and y = x − 4

a.8

3e.

1

3

b.2

3f.

1

9

c.1

12g.

9

2

d.1

2h.

5

6

____ 74. The area of the region bounded by y = sinx and y = cos x between x =π4

and x =5π4

is

a. 2 2 e. 2

b.2

2f. 4 2

c.3

2g. 2 3

d. 0 h. 3

____ 75. Find the area of the region bounded by the curves y = x3

− 2x and y = −x

a.1

4e.

1

2

b.1

9f.

1

6

c.1

18g. 4

d.2

15h. 2

____ 76. Find the area of the region bounded by the curves x = 4 − y2 and x = −3y

a.115

6e.

1

9

b.1

4f.

125

6

c.1

15g.

1

12

d. 20 h.43

2

Name: ________________________ ID: A

20

____ 77. The area of the region bounded by y2

= x, (y − 2)2

= x and the y-axis is

a.1

3e.

3

2

b.2

3f. 1

c.3

2g. 2

d. 0 h.1

2

____ 78. Find the area of the region bounded by the parabola x = y2 and the line x − 2y = 3.

a.29

3e.

41

3

b.32

3f.

44

3

c.35

3g.

47

3

d.38

3h.

50

3

____ 79. Find the area of the region bounded by the curve x = 4cos t, y = 3sin t, 0 ≤ t ≤ π , and the x-axis.

a. π e. π 2

b. 6π 2f. 12π 2

c. 3π g. 3π 2

d. 6π h. 12π

____ 80. Find the area of the region bounded by x = 1 + sin t, y = sin t, 0 ≤ t ≤ π , and the x-axis.

a. 4π e.π2

b.3π2

f.π6

c. 2π g. π

d.π4

h.π3

____ 81. Find the area of the region bounded by x = etcos t, y = e

tsin t, 0 ≤ t ≤ π , and the x-axis.

a.1

4e

2πe.

1

4(e

2π− 1)

b.1

2e

2πf.

1

2(e

2π− 1)

c.1

4(e

2π+ 1) g.

1

2(e

π− 1)

d.1

4e

πh. e

2π− 1

Name: ________________________ ID: A

21

____ 82. Find the volume of the solid obtained when the region bounded by the x-axis, the y-axis, and the line

y + x = 3 is rotated about the x-axis.

a. 3π e. 9πb. 2π f. 12πc. 8π g. 4πd. 10π h. 6π

____ 83. Find the volume of the solid obtained when the region bounded by the line y = 2x , the line x = 3, and the

x-axis is rotated about the y-axis.

a. 36π e. 24πb. 42π f. 30πc. 16π g. 48πd. 18π h. 27π

____ 84. Find the volume of the solid obtained when the region bounded by the curve y = sinx, 0 ≤ x ≤ π, and the

x-axis is rotated about the x-axis.

a.1

2π 2

e.π4

b.1

3π 2

f. π 2

c. π g.π3

d.π2

h.1

4π 2

____ 85. The base of a solid S is the parabolic region x,yÊËÁÁ ˆ

¯˜̃

|| y

2≤ x ≤ 1

ÏÌÓ

ÔÔÔÔÔÔ

¸˝˛

ÔÔÔÔÔÔ. Cross-sections perpendicular to the x-axis

are squares. Find the volume of S .

a. 1.5 e. 1.9

b. 1.6 f. 2.0

c. 1.7 g. 2.1

d. 1.8 h. 2.2

____ 86. Find the volume of the solid obtained by rotating about the line y = 1, the region bounded by

y = cos x, y = 0, x = 0, and x =π2

.

a. π e. π 2− 2π

b. π 2f. 2π −

1

4π 2

c. π 2−

π2

g. 2π −1

2π 2

d. π 2− π h. π −

1

4π 2

Name: ________________________ ID: A

22

____ 87. The volume of the solid obtained by rotating the region y = tanx, y = 0, and x =π4

about the x-axis is

a.4 − π

4e. ln( 2 − 1)

b.π(4 − π)

4f.

π(π − 4)

4

c. π ln( 2 − 1) g. π (4 − π)

d.π2

h.π4

____ 88. A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral

triangles. Find the volume of the solid.

a.π2

e.2π3

b.3

2f.

4 3

3

c.3π2

g.3π4

d.3 3

2h.

2 2

3

____ 89. Find the volume of the solid obtained when the region bounded by the curves y = x2

+ 1, y = 1, and x = 1 is

rotated about the line x =1.

a.π12

e.π3

b. 8π f.π6

c.π2

g. 4π

d.π4

h.10π

3

____ 90. Find the volume of the solid obtained when the region bounded by the curves y = x3, x = 1, and the x-axis is

rotated about the line x = −1.

a.127π

3e.

2π5

b.62π

3f.

π5

c.π2

g.9π10

d.π4

h.31π

3

Name: ________________________ ID: A

23

____ 91. The volume of the solid obtained by rotating the plane region enclosed by y = x4, y = 1, and x = 0 about the

y-axis is

a. π e.π3

b.2π3

f.4π3

c.π2

g.π6

d. 0 h. 1

____ 92. The volume of the solid obtained by rotating the plane region enclosed by y = x4, y = 1, and x = 0 about the x

= 1 is

a.7π15

e.14π

3

b.2π3

f.7π5

c.14π15

g.14π

5

d.14

15h.

7

15

____ 93. Find the volume of the solid obtained when the region bounded by the curve y = sinx, 0 ≤ x ≤ π, and the

x-axis is rotated about the y-axis.

a.1

2π 3

e. 2π 2

b. π 2f. 2π 3

c. π 3g.

1

2π 2

d. 4π 2h. 4π 3

____ 94. Find the volume of the solid obtained when the region above the x-axis, bounded by the x-axis and the curve

y = x − x3, is rotated about the y-axis.

a.2

7π e.

4

35π

b.8

35π f.

2

c.4

15π g.

4

d.8

15π h.

4

Name: ________________________ ID: A

24

____ 95. The volume of the solid obtained by rotating the plane region enclosed by y =sin(2x)

x, x =

π2

, the y-axis and

the x-axis about the y-axis is

a. 2π e. 4π

b. π f.π2

c. 2 g. 4

d. 1 h.1

12

____ 96. Find the average value of the function f(x) =x

x2

+ 9

, 0 ≤ x ≤ 4.

a. 0 e.1

2

b.2

5f. 5

c. 10 g. 4

d.1

5h.

4

5

____ 97. The density of a rod 9 meters long is x kg/m at a distance of x meters from one end of the rod. Find the

average density of the rod.

a. 6 e. 2

b.4

3f.

16

3

c. 3 g. 4

d. 1 h.8

3

____ 98. Find the average value of the function f(x) = sin3x on the interval 0,π3

È

Î

ÍÍÍÍÍÍÍÍ

˘

˚

˙̇˙̇˙̇˙̇.

a. 1 e.1

π

b.3

2f.

2

πc. 2 g. 0

d. 6 h.5

____ 99. Find the average value of the function f(x) = 4 − x2

on the interval −2,2ÈÎÍÍÍ

˘˚˙̇̇ .

a.π2

e. 2π

b. π f. 4πc. 8π g. 16πd. 3π h. 6π

Name: ________________________ ID: A

25

Short Answer

100. Evaluate the following integrals:

(a) x3

cos x4

+ 1ÊËÁÁÁ

ˆ¯˜̃̃ dx∫

(b)

cos xÊ

ËÁÁÁ

ˆ

¯˜̃˜

xdx∫

(c) cos 1 xÊ

ËÁÁ ˆ

¯˜̃

x2

dx∫

(d) cos arctanx( )

1 + x2

dx∫

101. Evaluate the following integrals:

(a) x3e

x4

+ 1Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜

dx∫

(b) e

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜

xdx∫

(c) e

1 xÊËÁÁÁ

ˆ¯˜̃̃

x2

dx∫

(d) e

arctan x( )

1 + x2

dx∫

Name: ________________________ ID: A

26

102. Evaluate the following integrals:

(a) xex

2

dx∫

(b) cos lnx( )

xdx∫

(c) sec

2x

tanxdx∫

(d) 3x

2

1 + x6

dx∫

103. Evaluate the following integrals:

(a) sin2x sinx dx0

π 4

(b) ex

sin exÊ

ËÁÁÁ

ˆ¯˜̃̃ dx∫

(c) x

x + 2dx

−1

2

(d) tanx dx∫

104. Evaluate the following integrals:

(a) x

x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

4dx∫

(b) 1

x lnx( )2

dxe

e2

(c) e

1 x

x2

dx∫

(d) x

x + 1dx

0

1

Name: ________________________ ID: A

27

105. Evaluate the following integrals:

(a) ex

+ e−xÊ

ËÁÁÁ

ˆ¯˜̃̃

2

dx∫

(b) x + 3

x2

+ 6xÊËÁÁÁ

ˆ¯˜̃̃

2dx∫

(c) x + 3

x2

+ 6xdx∫

(d) x cos x

1 + x4

dx− π 2

π 2

106. Evaluate the following integrals:

(a) u + u

3

udu∫

(b) 2x + 1| | dx−2

1

(c) t3

+ 13

⋅ t5

dt0

1

(d) 16 − x2

dx−4

4

Name: ________________________ ID: A

28

107. Evaluate the following integrals:

(a) sinx cos2x dx

0

π 2

(b) cos x

1 − sinxdx

0

π 2

(c) sinx ⋅ 16 − x

2

4 + x2

dx−4

4

(d) x3

+ 2 4 − x2

Ê

Ë

ÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜

dx−2

2

108. Evaluate the following integrals:

(a) x2

lnx dx∫

(b) x x + 1( )4

dx∫

(c) x2e

xdx∫

(d) x cos 2x( ) dx∫

109. Evaluate the following integrals:

(a) x sinx dx∫

(b) x sin x2Ê

ËÁÁÁ

ˆ¯˜̃̃ dx∫

(c) sin−1

x dx∫

(d) xe3x

dx∫

Name: ________________________ ID: A

29

110. Evaluate the following integrals:

(a) x sec2x dx∫

(b) 3x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃ ln x

2+ 1

ÊËÁÁÁ

ˆ¯˜̃̃ dx∫

(c) e

x

xdx∫

(d) ex

dx∫

111. Evaluate the following integrals:

(a) x cot−1

x dx∫

(b) lnx( )2

dx1

e

(c) x2

cos x dx∫

(d) cos lnx( ) dx∫

112. Determine a reduction formula for x lnx( )n

dx.1

e

113. (a) Use integration by parts to prove the reduction formula: xn

cos x dx = xn

sinx − n xn − 1

sinx dx∫∫

(b) Demonstrate your understanding of this formula by using it to evaluate: x4

cos x dx.∫

114. Let f be a twice differentiable function such that f(0) = 5, f(3) = 1, and f ′(3) = −2. Determine the value of

x ⋅ f ″ x( ) dx0

3

∫ .

Name: ________________________ ID: A

30

115. Evaluate the following integrals:

(a) sin4x cos x dx

0

π 2

(b) sin4x cos

3x dx

0

π 2

(c) sin2x cos42x dx

0

π 4

(d) cos4x dx

0

π

116. Evaluate the following integrals:

(a) x

x2

− x − 2dx∫

(b) 1

x x − 1( )2

dx∫

(c) 1

x3

+ xdx∫

(d) x

3+ x + 2

x2

+ 1dx∫

117. Evaluate the following integrals:

(a) lnx

x2

dx∫

(b) x

2+ x + 1

x2

− 1dx∫

(c) cos x

4 + sin2x

dx∫

(d) x3

4 − x2

dx∫

Name: ________________________ ID: A

31

118. Evaluate the following integrals:

(a) x + sinx( )2

dx∫

(b) xex

2+ 1

dx∫

(c) cos x sin3x dx

0

π 2

(d) 7x − 23

x2

− 7x + 12dx∫

119. Evaluate the following integrals:

(a) x x2

− 4 dx∫

(b) x2

− 4 dx∫

(c) 1

x2

+ 2x + 2dx∫

(d) x

3

1 + x2

dx∫

120. Evaluate the following integrals:

(a) x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃e

x3

+ 3xdx∫

(b) sin x dx∫

(c) x

2+ x + 1

xdx∫

(d) 4 + x2

dx∫

Name: ________________________ ID: A

32

121. Evaluate the following integrals:

(a) x

1 − x2

dx0

1 2

(b) ex + e

x

dx∫

(c) 3x + 2

x − 1dx∫

(d) ln 1 + x2Ê

ËÁÁÁ

ˆ¯˜̃̃ dx∫

122. Use Simpson’s Rule with n = 6 to approximate x3

+ 2 dx0

2

∫ .

123. Use Simpson’s Rule with n = 4 to approximate 1

1 + x2

dx0

1

∫ .

124. Use the Midpoint Rule with 2 equal subdivisions to get an approximation for ln 5.

125. Use (a) the Trapezoidal Rule with n = 8 and (b) Simpson's Rule with n = 8 to approximate ex

0

2

∫ dx. Round

your answers to six decimal places.

126. Estimate 1

x2

dx1

3

∫ using the Trapezoidal Rule with n = 4. Then use the error bound ET||

|| ≤

k b − a( )3

12n2

to

estimate the accuracy.

127. Estimate 1

x2

dx1

3

∫ using the Midpoint Rule with n = 4. Then use the error bound E M||

|| ≤

k b − a( )3

24n2

to

estimate the accuracy.

128. Estimate 1

x2

dx1

3

∫ using Simpson’s Rule with n = 4. Then use the error bound ES||

|| ≤

k b − a( )5

180n4

to estimate

the accuracy.

129. Consider the integral cos x3Ê

ËÁÁÁ

ˆ¯˜̃̃

0

0.5

∫ dx. Approximating it by the Midpoint Rule with n equal subintervals, give

an estimate for n which guarantees that the error is bounded by 1

104

.

Name: ________________________ ID: A

33

130. (a) Estimate ln2 =1

x1

2

∫ dx using Simpson's Rule with n = 4.

(b) Estimate the error of the approximation in part (a).

(c) How large should we take n to guarantee that the estimate by Simpson's Rule is accurate to within 0.001?

131. Two students use Simpson's Rule to estimate x2

+ 4x − 5ÊËÁÁÁ

ˆ¯˜̃̃ dx

0

1

∫ . One divides the interval into 30 equal

subintervals and the other into 60 equal subintervals. How will the accuracy of their estimates compare?

132. Two students use Simpson's Rule to estimate x3

+ 2x2

− x + 1ÊËÁÁÁ

ˆ¯˜̃̃ dx

1

2

∫ . One divides the interval into 30 equal

subintervals and the other into 60 equal subintervals. How will the accuracy of their estimates compare?

133. The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in

the figure. Use Simpson's Rule to estimate the area of the pool.

134. Below is a table of values for a continuous function f.

x −1 −0.5 0 0.5 1

f x( ) 2.0 1.5 −1.0 0.5 −3.0

(a) Use the Trapezoidal Rule with n = 4 to approximate f x( )−1

1

∫ dx.

(b) Use Simpson's Rule with n = 4 to approximate f x( )−1

1

∫ dx.

Name: ________________________ ID: A

34

135. The following table shows the speedometer readings of a truck, taken at ten minute intervals during one hour

of a trip.

Time (min) 0 10 20 30 40 50 60

Speed (mi/h) 40 45 50 60 70 65 60

Use the table and the indicated technique to estimate the distance that the truck traveled in the hour.

(a) The Trapezoidal Rule

(b) The Midpoint Rule

(c) Simpson's Rule

136. Intelligence Quotient (IQ) scores are assumed to be normally distributed in the population. The probability

that a person selected at random from the general population will have an IQ between 100 and 120 is given

by p x( )0

2

∫ dx. Use the graph of p (x) graphed below to answer the questions which follow:

(a) Use Simpson's Rule with n = 4 to approximate p x( )0

2

∫ dx.

(b) The probability that a person selected from the general population will have an IQ score between 80 and

120 is given by p x( )−2

2

∫ dx. What is the approximate value of p x( )−2

2

∫ dx?

(c) Since p (x) represents a probability distribution, the entire area of the region under the graph is exactly 1.

Using this information, what is the approximate probability that a person selected at random from the

general population will have an IQ score over 120?

Name: ________________________ ID: A

35

137. A scientist collects the following data and plots it in the coordinate plane.

x 2 2.5 3 3.5 4 4.5 5

y 4 10 8 6 14 10 12

(a) Use Simpson's rule with n = 6 to estimate the area under the graph of a continuous function drawn

through these points.

(b) If it is known that −4 ≤ f4( )

x( ) ≤ 1 for all x, estimate the error involved in the approximation in part (a).

(c) How large do we have to choose n so that the approximation S n (Simpson's Rule) to the integral is

accurate to within 0.001?

138. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

(a) 1

x2

+ 4−∞

2

∫ dx

(b) 1

x1.0001

1

∫ dx

(c) 1

x0.9999

1

∫ dx

139. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

(a) 1

x3 23

∫ dx

(b) 1

x lnx( )e

∫ dx

(c) x

x2

+ 5ÊËÁÁÁ

ˆ¯˜̃̃

20

∫ dx

Name: ________________________ ID: A

36

140. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

(a) x− 5 4

1

∫ dx

(b) 1

x−∞

1

∫ dx

(c) xe−2x

0

∫ dx

141. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

(a) 1

x − 1( )2

0

1

∫ dx

(b) 1

1 − x0

1

∫ dx

(c) 1

x − 2( )2 30

4

∫ dx

142. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

(a) 1

x − 10

2

∫ dx

(b) 1

x lnx( )2

2

∫ dx

(c) lnx

x0

1

∫ dx

Name: ________________________ ID: A

37

143. Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

(a) 1

x − 1( )2

0

2

∫ dx

(b) dx

x − 1( )2 30

3

∫ dx

(c) sec2t

0

π

∫ dt

144. Use the Comparison Theorem to determine whether sin

4x

x2

1

∫ dx is convergent or divergent. Justify your

answer.

145. Use the Comparison Theorem to determine whether 1

x + ex

0

∫ dx is convergent or divergent. Justify your

answer.

146. Use the Comparison Theorem to determine whether dx

4x2

+ 12

∫ is convergent or divergent. Justify your

answer.

147. Use the Comparison Theorem to determine whether dx

4x2

− 12

∫ is convergent or divergent. Justify your

answer.

148. Use the Comparison Theorem to determine whether 1

x + 10

∫ dx is convergent or divergent. Justify your

answer.

149. The area of the region under the f x( ) =1

4 + x2

is defined to be f x( )−∞

∫ dx. Find this area.

Name: ________________________ ID: A

38

150. The work done against gravity in propelling an object with mass m kg to an altitude of h m above the surface

of the earth is given by

W =GME m

r2

6.37 x 106

6.37 x 106

+ h

∫ dr

where 6.37 × 106 m is the radius of the earth, G ≈ 6.667 × 10−11

N ⋅ m2

/ kg2, and M E ≈ 5.90 × 10

24kg is the

mass of the earth.

(a) Find the work required to launch a 1000-kilogram satellite vertically to an altitude of 1000 km.

(b) The formula shows that the work is dependent on h. Show that GME m

r2

6.37 × 106

∫ dr is convergent. What

is the physical significance of the fact that this improper integral is finite?

151. At the Dr. T. Bottling Company, the amount of soda dispensed into the cans is normally distributed so the

probability that a can will contain a certain amount of soda may be calculated using the area under the curve

y = p (x), where p(x) = 1

2πe

−x2

2. Use your graphing calculator to produce a graph of this function.

(a) The probability that a can will contain between 12.0 and 12.2 ounces of soda is given by

1

2πe

−x2

2dx.

−0.22

0.22

∫ Use Simpson's Rule with n = 4 to approximate this probability.

(b) The probability that a can will contain at least 12 ounces of soda is given by 1

2πe

−x2

2dx.

−0.22

∫ What

is the approximate value of this integral?

152. Find the area of the region bounded by the curves y = x3

+ x2and y = 2x

2+ 2x .

153. Find the area of the region bounded by the curves y = (x − 2)2

− 1 and y = 3 − x .

154. Find the area of the region bounded by the curves y = lnx, x = e and the x-axis.

155. Find the area of the region bounded by the curves x = 4 − y2 and x = y

2− 4.

156. Find the area of the region bounded by the curves x = y2

− 7 and x = y − 1.

157. Let R be the region bounded by: y = x3, the tangent to y = x

3at 1,1Ê

ËÁÁ ˆ

¯˜̃ , and the x-axis.

Find the area of R integrating

(a) with respect to x.

(b) with respect to y.

Name: ________________________ ID: A

39

158. Find the area of the region bounded by the curves f(x) =7

2x + 1 and g(x) = 2

x.

159. Using the help of a graphing calculator, find the area of the region bounded by the curves y = x(x − 1) and

y = sin(x).

160. Find the area of the region bounded by the curves f(x) = 3 ⋅ sin(π4

x), g(x) = 2x

− 1 and x ≥ 0.

161. Find the area of the shaded region:

162. Find the area of the shaded region:

Name: ________________________ ID: A

40

163. Find the area of the shaded region:

164. Find the area of the shaded region:

165. Find the area of the region bounded by y = x(x − 1)(x − 2) and the x − axis.

166. Find the area of the region bounded by y = x3

− 6x and y = −2x.

167. A particle is moving in a straight line and its velocity is given by v(t) = 3t2

− 12t + 9, where t is measure in

seconds and v in meters per second. Find the distance traveled by the particle during the time interval 0,5ÈÎÍÍÍ

˘˚˙̇̇ .

168. A stone is thrown straight up from the top of a tower that is 80 ft tall with initial velocity 64 ft/s. What is the

total distance traveled by the stone when it hits the ground?

Name: ________________________ ID: A

41

169. Express the area of the given region as a definite integral. Do not evaluate.

170. Express the area of the given region as a definite integral. Do not evaluate.

171. Find the volume of the solid obtained by rotating the region bounded by the curves

y = x − 2 , y = 0, and x = 6 about the x-axis.

172. Find the volume of the solid obtained by rotating the region bounded by the curves y = 2x2, x = 0, and y = 2

about the y-axis.

173. Consider the region in the xy-plane between x = 0, x =π2

, bounded by y = 0 and y = sinx . Find the volume

of the solid generated by rotating this region about the x-axis.

174. Find the volume of the solid formed when the region bounded by the curves y = x3

+ 1, x = 1, and y = 0 is

rotated about the x-axis.

Name: ________________________ ID: A

42

175. Find the volume of the solid generated by rotating about the line y = −1 the region bounded by the graphs of

the equations y = x2

− 4x + 5 and y = 5 − x.

176. Find the volume of the solid obtained by rotating the region bounded by the curves y = 3 − x2

and y = 2

about the line y = 2.

177. Find the volume of the solid obtained by rotating the region bounded by the curves

y = 9 − x2

, x = 0, and y = 1 about the x-axis.

178. Find the volume of the solid obtained by rotating the region bounded by the curves y = x3, y = 0, and x = 1

about the line x = 2.

179. Find the volume of the solid obtained by rotating the region bounded by the curve y = 1 − x2

and the

x-axis about the line y = 2.

180. Find the volume of the solid generated by rotating the region bounded by y =1

x, x = 1, and the x-axis about

the x-axis.

181. The base of a certain solid is a plane region R enclosed by the x-axis and the curve y = 1 − x2. Each

cross-section of the solid perpendicular to the y-axis is an isosceles triangle of height 4 with its base lying in

R. Find the volume of the solid.

182. The base of a certain solid is a plane region R enclosed by the x-axis and the curve y = 1 − x2. Each

cross-section of the solid perpendicular to the y-axis is an equilateral triangle with its base lying in R. Find

the volume of the solid.

183. The base of a certain solid is a plane region R enclosed by the x-axis and the curve y = 1 − x2. Each

cross-section of the solid perpendicular to the y-axis is an isosceles right triangle with hypotenuse lying in R.

Find the volume of the solid.

184. The base of a certain solid is the triangular region with vertices (0,0), (1,1), and (2,0). Cross-sections

perpendicular to the x-axis are semicircles. Find the volume of the solid.

185. The base of a certain solid is an elliptical region with boundary curve x

2

16+

y2

9= 1. Cross-sections

perpendicular to the x-axis are squares. Find the volume of the solid.

186. Find the volume of the solid obtained by rotating the region bounded by xy = 1, y = 1, y = 2 and the y-axis

about the x-axis.

Name: ________________________ ID: A

43

187. Find the volume of the solid obtained by rotating the region bounded by y =3

x2

+ 9

, x = 4, the y-axis and

the x-axis about the y-axis.

188. Find the volume of the solid obtained by rotating the region bounded by y = 2x − x2 and the x-axis about the

line x = 2.

189. Find the volume of the solid obtained by rotating the region bounded by y = x3

, y = e−2x

and x = 1 about

the line x = 1. (Use a graphing calculator.)

190. Let R be region bounded by the graph of f(x) = x2, g(x) =

1

x, and the line y = 3.

(a) Find the volume of the solid obtained by rotating R about the x-axis.

(b) Find the volume of the solid obtained by rotating R about the y-axis.

(c) Find the volume of the solid obtained by rotating R about the line y = 3.

(d) Find the volume of the solid obtained by rotating R about the line x = 2.

191. Let R be region bounded by the curve 4y = x2, x = 2y − 4.

(a) Find the volume of the solid obtained by rotating R about the x-axis.

(b) Find the volume of the solid obtained by rotating R about the line x = 5.

(c) Find the volume of the solid obtained by rotating R about the line y = –1.

192. The region R is given by the shaded area in the figure below:

(a) Find the area of the shaded region R.

(b) Find the volume of the solid obtained by rotating R about

(i) the x-axis. (ii) the y-axis. (iii) the line x = 2. (iv) the line y = 4

Name: ________________________ ID: A

44

193. A hole of radius 6 cm is drilled through the center of a sphere of radius 10 cm. How much of the ball’s

volume is removed?

194. Let f(x) = (3 − x)(3 + x) , find c such that f ave = f(c) on the interval 0,3ÈÎÍÍÍ

˘˚˙̇̇ .

195. Let f(x) = ln x , find c such that f ave = f(c) on the interval 1,eÈÎÍÍÍ

˘˚˙̇̇ .

196. Find the average value of the function whose graph is given below.

197. Find the average value of the function whose graph is given below.

198. Estimate the average value of the function whose graph is given below.

Name: ________________________ ID: A

45

199. Find the average value of f(x) = x 25 − x2

on the interval 0,5ÈÎÍÍÍ

˘˚˙̇̇ . At how many points in the interval does

f(x) have this value?

200. The temperature (in °F) in a certain city t hours after 9 A.M. is approximated by the function

T(t) = 50 + 14sin(π12

t). Find the average temperature during the period from 9 A.M. to 9 P.M..

201. The temperature (in °C) of a metal rod 5 m long is 4x at a distance x meters from one end of the rod. What is

the average temperature of the rod?

202. A stone is dropped from a bell tower 100 feet tall. Find the average velocity of the stone from the instant it is

dropped until it strikes the ground. (Assume that the acceleration due to gravity is 32 ft/s2.)

203. A culture of bacteria is doubling every hour. What is the average population over the first two hours if we

assume that the culture initially contained two million organisms?

204. Find the average value of f(x) = 6 − x| | on the interval −2,2ÈÎÍÍÍ

˘˚˙̇̇ .

205. A particle is moving along a straight line so that its velocity at time t is v(t) = 3t2. At what time t during the

interval 0 < t < 3 is its velocity the same as its average velocity over the entire interval?

206. The following table shows the velocity of a car (in mi/hr) during the first five seconds of a race.

t (s) 0 1 2 3 4 5

v (mi/h ) 0 20 32 46 54 62

Determine the average velocity of the car during this five-second interval.

Name: ________________________ ID: A

46

207. The graph of a continuous function g(x) is given below:

List from smallest to largest:

(a) The average value of g over 0,10ÈÎÍÍÍ

˘˚˙̇̇ (d) g(x)

3

5

∫ dx

(b) The average rate of change of g over 0,10ÈÎÍÍÍ

˘˚˙̇̇ (e) g(x)

0

10

∫ dx

(c) g ′(6) (f) g(x)3

6

∫ dx

208. Consider the region R bounded by y =1

2x , y = 1, and the y-axis.

(a) Find the area of R.

(b) Find the average height of R.

(c) Find the volume, V, of the solid obtained by rotating R about the x-axis.

(d) A cross section of the solid generated by part (c) taken perpendicular to the x-axis is a washer. Determine

the average area of the cross sections of the solid.

209. The voltage (in volts) at an electrical outlet is a function of time (in seconds) given by V(t) = V0 cos(120πt)

where V0 is a constant representing the maximum voltage.

(a) What is the average value of the voltage over one second?

(b) How many times does the voltage reach a maximum in one second?

(c) Define the new function S(t) = (V (t))2. Compute S , the average value of S(t) over one cycle.

(d) Instead of the average voltage, engineers use the root mean square Vrms= S . Determine Vrms in terms

of V0.

(e) The standard household voltage in the United Stated is 100 volts. This means that Vrms = 110. What is the

value of V0?

ID: A

1

104ex1questions

Answer Section

MULTIPLE CHOICE

1. ANS: B PTS: 1

2. ANS: G PTS: 1

3. ANS: E PTS: 1

4. ANS: D PTS: 1

5. ANS: B PTS: 1

6. ANS: B PTS: 1

7. ANS: D PTS: 1

8. ANS: C PTS: 1

9. ANS: H PTS: 1

10. ANS: C PTS: 1

11. ANS: F PTS: 1

12. ANS: D PTS: 1

13. ANS: G PTS: 1

14. ANS: E PTS: 1

15. ANS: E PTS: 1

16. ANS: F PTS: 1

17. ANS: C PTS: 1

18. ANS: F PTS: 1

19. ANS: A PTS: 1

20. ANS: H PTS: 1

21. ANS: G PTS: 1

22. ANS: E PTS: 1

23. ANS: A PTS: 1

24. ANS: C PTS: 1

25. ANS: F PTS: 1

26. ANS: C PTS: 1

27. ANS: D PTS: 1

28. ANS: E PTS: 1

29. ANS: H PTS: 1

30. ANS: C PTS: 1

31. ANS: C PTS: 1

32. ANS: G PTS: 1

33. ANS: B PTS: 1

34. ANS: B PTS: 1

35. ANS: C PTS: 1

36. ANS: E PTS: 1

37. ANS: B PTS: 1

38. ANS: A PTS: 1

39. ANS: A PTS: 1

ID: A

2

40. ANS: G PTS: 1

41. ANS: E PTS: 1

42. ANS: D PTS: 1

43. ANS: G PTS: 1

44. ANS: F PTS: 1

45. ANS: A PTS: 1

46. ANS: C PTS: 1

47. ANS: F PTS: 1

48. ANS: A PTS: 1

49. ANS: E PTS: 1

50. ANS: E PTS: 1

51. ANS: C PTS: 1

52. ANS: G PTS: 1

53. ANS: G PTS: 1

54. ANS: E PTS: 1

55. ANS: D PTS: 1

56. ANS: B PTS: 1

57. ANS: B PTS: 1

58. ANS: H PTS: 1

59. ANS: H PTS: 1

60. ANS: D PTS: 1

61. ANS: H PTS: 1

62. ANS: H PTS: 1

63. ANS: C PTS: 1

64. ANS: C PTS: 1

65. ANS: B PTS: 1

66. ANS: F PTS: 1

67. ANS: A PTS: 1

68. ANS: H PTS: 1

69. ANS: H PTS: 1

70. ANS: H PTS: 1

71. ANS: F PTS: 1

72. ANS: A PTS: 1

73. ANS: G PTS: 1

74. ANS: A PTS: 1

75. ANS: E PTS: 1

76. ANS: F PTS: 1

77. ANS: B PTS: 1

78. ANS: B PTS: 1

79. ANS: D PTS: 1

80. ANS: E PTS: 1

81. ANS: C PTS: 1

82. ANS: E PTS: 1

83. ANS: A PTS: 1

84. ANS: A PTS: 1

ID: A

3

85. ANS: F PTS: 1

86. ANS: F PTS: 1

87. ANS: B PTS: 1

88. ANS: F PTS: 1

89. ANS: F PTS: 1

90. ANS: G PTS: 1

91. ANS: B PTS: 1

92. ANS: C PTS: 1

93. ANS: E PTS: 1

94. ANS: C PTS: 1

95. ANS: A PTS: 1

96. ANS: E PTS: 1

97. ANS: E PTS: 1

98. ANS: F PTS: 1

99. ANS: A PTS: 1

SHORT ANSWER

100. ANS:

(a) 1

4sin x

4+ 1

ÊËÁÁÁ

ˆ¯˜̃̃ + C

(b) 2sin xÊ

ËÁÁÁ

ˆ

¯˜̃˜ + C

(c) −sin 1 xÊËÁÁ ˆ

¯˜̃ + C

(d) sin arctanx( ) + C

PTS: 1

101. ANS:

(a) 1

4e

x4

+ 1Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜

+ C

(b) 2ex

Ê

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜

+ C

(c) −e1 x

ÊËÁÁÁ

ˆ¯˜̃̃

+ C

(d) earctan x( )

+ C

PTS: 1

ID: A

4

102. ANS:

(a) 1

2e

x2

+ C

(b) sin lnx( ) + C

(c) ln tanx| | + C

(d) tan−1

x3Ê

ËÁÁÁ

ˆ¯˜̃̃ + C

PTS: 1

103. ANS:

(a) 2

6

(b) −cos ex

+ C

(c) 2

3

(d) ln sec x| | + C

PTS: 1

104. ANS:

(a) −1

6 x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃

3+ C

(b) 1

2

(c) −e1 x

+ C

(d) 1 − ln2

PTS: 1

105. ANS:

(a) 1

2e

2x−

1

2e

−2x+ 2x + C

(b) −1

2 x2

+ 6xÊËÁÁÁ

ˆ¯˜̃̃

+ C

(c) 1

2ln x

2+ 6x

||

|| + C

(d) 0

PTS: 1

ID: A

5

106. ANS:

(a) 2 u + 3 u3

+ C

(b) 9

2

(c) 2

3

14+

3

28

(d) 8π

PTS: 1

107. ANS:

(a) 1

3

(b) 2

(c) 0

(d) 4π

PTS: 1

108. ANS:

(a) 1

3x

3lnx −

1

9x

3+ C

(b) 1

5x x + 1( )

5−

1

30x + 1( )

6+ C

(c) x2e

x− 2xe

x+ 2e

x+ C

(d) x

2sin 2x( ) +

1

4cos 2x( ) + C

PTS: 1

109. ANS:

(a) −x cos x + sinx + C

(b)

−cos x2Ê

ËÁÁÁ

ˆ¯˜̃̃

2+ C

(c) x sin−1

x + 1 − x2

+ C

(d) xe

3x

3−

e3x

9+ C

PTS: 1

ID: A

6

110. ANS:

(a) ln cos x| | + x tanx + C

(b) x3

+ xÊËÁÁÁ

ˆ¯˜̃̃ ln x

2+ 1

ÊËÁÁÁ

ˆ¯˜̃̃ −

2x3

3+ C

(c) 2ex

+ C

(d) 2ex

x − 1Ê

ËÁÁÁ

ˆ

¯˜̃˜ + C

PTS: 1

111. ANS:

(a) x

2

2cot

−1x +

x

2−

1

2tan

−1x + C

(b) e − 2

(c) 2x cos x + x2

sinx − 2sinx + C

(d) x

2cos lnx( ) + sin lnx( )ÊËÁÁ ˆ

¯˜̃ + C

PTS: 1

112. ANS:

In =e

2

2−

n

2In − 1 ,where In − 1 = x lnx( )

n − 1dx

1

e

PTS: 1

113. ANS:

(a) u = xn,dv = cos x dx

(b) 4x3

− 24xÊËÁÁÁ

ˆ¯˜̃̃ cos x + x

4− 12x

2+ 24

ÊËÁÁÁ

ˆ¯˜̃̃ sinx + C

PTS: 1

114. ANS: −2

PTS: 1

115. ANS:

(a) 1

5

(b) 2

35

(c) 1

10

(d) 3π8

PTS: 1

ID: A

7

116. ANS:

(a) 1

3ln x − 2( )

2x + 1( )

||

|| + C

(b) lnx

x − 1

|||

|||−

1

x − 1+ C

(c) ln x| | − ln x2

+ 1Ê

Ë

ÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜

+ C

(d) x

2

2+ 2arctan x + C

PTS: 1

117. ANS:

(a) −lnx

x−

1

x+ C

(b) x +1

2ln

x − 1| |3

x + 1| |

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜̃˜

+ C

(c) 1

2tan

−1 sinx

2

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃ + C

(d) −1

153x

2+ 8

ÊËÁÁÁ

ˆ¯˜̃̃ 4 − x

2ÊËÁÁÁ

ˆ¯˜̃̃

3 2

+ C

PTS: 1

118. ANS:

(a) −1

4sin2x − 2x cos x + 2sinx +

x3

3+

x

2+ C

(b) 1

2e

x2

+ 1+ C

(c) 1

4

(d) ln x − 3( )2

x − 4| |5Ê

ËÁÁÁ

ˆ

¯˜̃˜ + C

PTS: 1

119. ANS:

(a) 1

3x

2− 4

ÊËÁÁÁ

ˆ¯˜̃̃

3 2

+ C

(b) −2ln x2

− 4 + xÊ

Ë

ÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜

+x x

2− 4

2+ C

(c) tan−1

x + 1( ) + C

(d) 1

2x

2− ln x

2+ 1

ÊËÁÁÁ

ˆ¯˜̃̃

È

Î

ÍÍÍÍÍÍ

˘

˚

˙̇˙̇˙̇ + C

PTS: 1

ID: A

8

120. ANS:

(a) 1

3e

x3

+ 3xÊ

Ë

ÁÁÁÁÁÁÁ

ˆ

¯

˜̃̃˜̃̃˜

+ C

(b) −2 x cos x + 2sin x + C

(c) 2

5x

5 2+

2

3x

3 2+ 2 x + C

(d) 2ln x2

+ 4 + x|||

|||+

x x2

+ 4

2+ C

PTS: 1

121. ANS:

(a) 1 −3

2

(b) ee

x

+ C

(c) 5ln x − 1| | + 3x + C

(d) x ln x2

+ 1ÊËÁÁÁ

ˆ¯˜̃̃ + 2tan

−1x − 2x + C

PTS: 1

122. ANS: ≈ 3.86

PTS: 1

123. ANS: ≈ 0.785

PTS: 1

124. ANS:

ln5 ≈ 1.5

PTS: 1

125. ANS:

(a) T = 6.422298

(b) S = 6.389194

PTS: 1

126. ANS:

ET||

|| ≤

1

4

PTS: 1

127. ANS:

E M||

|| ≤

1

8

PTS: 1

ID: A

9

128. ANS:

ES||

|| ≤

1

12

PTS: 1

129. ANS:

n > 15 (answers may vary)

PTS: 1

130. ANS:

(a) 0.693

(b) 1

1920

(c) n = 4

PTS: 1

131. ANS:

Since f x( ) = x2

+ 4x − 5, the graph of f is a parabola. Therefore, Simpson's Rule gives an exact answer for

the integral.

PTS: 1

132. ANS:

Since f4( )

x( ) = 0, Es||

|| = 0, therefore using Simpson's Rule gives an exact answer for the integral.

PTS: 1

133. ANS:

84.26 m2

PTS: 1

134. ANS:

(a) 1

4

(b) 5

6

PTS: 1

135. ANS:

(a) 170

3miles

(b) 170

3miles

(c) 170

3miles

PTS: 1

ID: A

10

136. ANS:

(a) ≈ 0.47

(b) ≈ 0.94

(c) ≈ 0.03

PTS: 1

137. ANS:

(a) 82

3

(b) 1

240

(c) n ≥ 9

PTS: 1

138. ANS:

(a) 3π4

(b) 10,000

(c) Divergent

PTS: 1

139. ANS:

(a) 2

3

(b) Divergent

(c) 1

10

PTS: 1

140. ANS:

(a) 4

(b) Divergent

(c) 1

4

PTS: 1

141. ANS:

(a) Divergent

(b) 2

(c) 6 23

PTS: 1

ID: A

11

142. ANS:

(a) Divergent

(b) 1

ln2

(c) −4

PTS: 1

143. ANS:

(a) Divergent

(b) 3 23

+ 3

(c) Divergent

PTS: 1

144. ANS:

sin4x ≤ 1 ⇒

sin4x

x2

≤1

x2

. We know1

x2

dx1

∫ is convergent, thus sin

4x

x2

1

∫ dx is convergent.

PTS: 1

145. ANS:

1

x + ex

≤1

ex

= e−x

. e−x

0

∫ dx = 1 is convergent, thus 1

x + ex

0

∫ dx is convergent.

PTS: 1

146. ANS:

Divergent. Answers will vary.

PTS: 1

147. ANS:

Divergent. Answers will vary.

PTS: 1

148. ANS:

Divergent. Answers will vary.

PTS: 1

149. ANS:

π2

PTS: 1

150. ANS:

(a) About 8.3787 × 109 J

(b) It is possible to launch an object out of the earth's gravitational field.

PTS: 1

ID: A

12

151. ANS:

(a) About 0.17

(b) About 0.585

PTS: 1

152. ANS:

37

12

PTS: 1

153. ANS:

9

2

PTS: 1

154. ANS:

lnx dx = 11

e

PTS: 1

155. ANS:

64

3

PTS: 1

156. ANS:

125

6

PTS: 1

157. ANS:

(a) 1

12

(b) 1

12

PTS: 1

158. ANS:

About 10.43

PTS: 1

159. ANS:

sin (x) − x x − 1( )0

A

∫ dx ≈ 0.944 where A ≈ 1.618.

PTS: 1

ID: A

13

160. ANS:

2 +12

π−

3

ln2

PTS: 1

161. ANS:

9

43

3− ln3 −

3

4

PTS: 1

162. ANS:

10

3

PTS: 1

163. ANS:

9

2

PTS: 1

164. ANS:

9

PTS: 1

165. ANS:

1

2

PTS: 1

166. ANS:

8

PTS: 1

167. ANS:

28 m

PTS: 1

168. ANS:

208 feet

PTS: 1

ID: A

14

169. ANS:

3 −1

4x

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃0

2

∫ − 3 − x( ) dx + 3 −1

4x

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃2

4

∫ −x

2dx OR

2y − 3 − yÊËÁÁ ˆ

¯˜̃

1

2

∫ dy + 12 − 4yÊËÁÁ ˆ

¯˜̃

2

3

∫ − 3 − yÊËÁÁ ˆ

¯˜̃dy

PTS: 1

170. ANS:

2 + 4 + xÊ

ËÁÁÁ

ˆ

¯˜̃˜

−4

−3

∫ − 2 − 4 + xÊ

ËÁÁÁ

ˆ

¯˜̃˜ dx + 1 − 1 − x

Ê

ËÁÁÁ

ˆ

¯˜̃˜

−3

0

∫ − 2 − 4 + xÊ

ËÁÁÁ

ˆ

¯˜̃˜ dx OR

00

2

∫ − y2

− 4yÊËÁÁÁ

ˆ¯˜̃̃ dy + 2y − y

2ÊËÁÁÁ

ˆ¯˜̃̃

2

3

∫ − y2

− 4yÊËÁÁÁ

ˆ¯˜̃̃ dy

PTS: 1

171. ANS:

PTS: 1

172. ANS: π

PTS: 1

173. ANS:

π 2

4

PTS: 1

174. ANS:

23π14

PTS: 1

175. ANS:

162π5

PTS: 1

176. ANS:

16π15

PTS: 1

ID: A

15

177. ANS:

32 2π

3

PTS: 1

178. ANS:

3π5

PTS: 1

179. ANS:

2π 2−

4π3

PTS: 1

180. ANS: π

PTS: 1

181. ANS:

8

3

PTS: 1

182. ANS:

3

2

PTS: 1

183. ANS:

1

2

PTS: 1

184. ANS:

π12

PTS: 1

185. ANS:

192

PTS: 1

ID: A

16

186. ANS:

2π y1

y− 0

Ê

Ë

ÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃1

2

∫ dy = 2π

PTS: 1

187. ANS:

2π x3

x2

+ 9

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜̃̃0

4

∫ dx = 12π

PTS: 1

188. ANS:

2π (2 − x)(2x − x2

0

2

∫ ) dx =8π3

PTS: 1

189. ANS:

2π (1 − x)( x3

− e−2x

A

1

∫ ) dx ≈ 0.695 where A ≈ 0.239

PTS: 1

190. ANS:

(a) π5

36 3 − 24Ê

ËÁÁÁ

ˆ

¯˜̃˜

(b) 10π

3

(c) π24 3

5+

4

5− 6ln3

Ê

Ë

ÁÁÁÁÁÁÁÁÁ

ˆ

¯

˜̃˜̃˜̃˜̃̃

(d) π 8 3 − 6 − 4ln3Ê

ËÁÁÁ

ˆ

¯˜̃˜

PTS: 1

191. ANS:

(a) 144π

5

(b) 72π

(c) 234π

5

PTS: 1

ID: A

17

192. ANS:

(a) 7

3

(b) (i) 31

5π (ii)

15

2π (iii)

11

6π (iv)

187

15π

PTS: 1

193. ANS:

1952π3

PTS: 1

194. ANS:

1

3(3 − x)(3 + x)

0

3

∫ dx = 6 = (3 − c)(3 + c) = 9 − c2 ⇒ 3

PTS: 1

195. ANS:

1

e − 1lnx

1

e

∫ dx =1

e − 1= lnc ⇒ c = e

1

e − 1≈ 1.790

PTS: 1

196. ANS:

0

PTS: 1

197. ANS:

3

4

PTS: 1

198. ANS:

2

PTS: 1

199. ANS:

25

3, 2

PTS: 1

200. ANS:

59° F

PTS: 1

ID: A

18

201. ANS:

10° C

PTS: 1

202. ANS:

40 ft/s

PTS: 1

203. ANS:

3

ln2 million ≈ 4,328,085

PTS: 1

204. ANS:

5

PTS: 1

205. ANS:

t = 3

PTS: 1

206. ANS:

Answer may vary between 30.4 mph and 42.8 mph.

PTS: 1

207. ANS:

(f) < (d) < (b) < (a) < (c) < (e)

PTS: 1

208. ANS:

(a) 4

3

(b) 1

3

(c) 2π

(d) π2

PTS: 1

ID: A

19

209. ANS:

(a) 0

(b) 61 for 0 ≤ t ≤ 1

(c) 1

2V

0

2

(d) Vrms =1

2V0

(e) V0 = 2 ⋅ 110 ≈ 156 volts

PTS: 1