chapter 7: applications of integration -...

Chapter 7: Applications of Integration

1. Find the area of the region bounded by the equations by integrating (i) with respect to x and (ii) with respect to y.

2366

x yx y= −= −

A) 1099

6A = B) 2197

6A = C) 2197

12A = D) 1099

12A = E) 2195

12A =

2. Find the area of the region bounded by equations by integrating (i) with respect to x and

(ii) with respect to y.

2

72y xy x== −

A) 1637

4A = B) 819

2A = C) 4913

12A = D) 819

4A = E) 4913

6A =

3. Find the area of the region bounded by the graphs of the algebraic functions.

2( ) 14

( ) 0f x xg x

= −=

x

A) 229

2A = B) 686

3A = C) 1373

6A = D) 1372

3A = E) 344

3A =


2( ) 12 36

( ) 8( 6)f x x xg x x

= + += +

A) 256

3A = B) 512

3A = C) 128

3A = D) 292

3A = E) 364

3A =


3( ) 14

( ) 14f x xg x x

= −= −

A) 1

42A = B) 1

29A = C) 1

4A = D) 27

28A = E) 41

42A =

Copyright © Houghton Mifflin Company. All rights reserved. 207



2( ) 12, ( ) 0, –12, 13f y y g y y y= + = = = A) 3097

6A = B) 4825

3A = C) 3097

3A = D) 2414

3A = E) 4828

3A =

7. Find the area of the region bounded by the graphs of the equations.

2

16( ) , 0, 0 81xf x y x

x= = ≤

+≤

A) 8ln(65)A = D) ( )63ln 8A = B) 8ln(63)A = E) None of the above C) ( )65ln 8A =


( ) sin(2 ), ( ) cos( ),2 6

f x x g x x xπ π= = − ≤ ≤

A)

3/ 238

A = B) 3/ 232

A = C) 3/ 298

A = D) 3/ 292

A = E) None of the above


2

( ) , 0, 0 1xf x xe y x−= = ≤ ≤ A) 1

4eA −

= B) 11

4eA−−

= C) 13eA −

= D) 12eAe−

= E) 14eA +

=

208 Copyright © Houghton Mifflin Company. All rights reserved.


10. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

2

6, 1212xy y= = −

A) 22 6 2

6 2

201612 36 212 5xV dxπ π

−

= − − =

∫

B) 22 6 2

6 2

403212 36 212 5xV dxπ π

−

= − − =

∫

C) 22 12

12

403212 36 212 5xV dxπ π

−

= − − =

∫

D) 22 12

12

201612 36 212 5xV dxπ π

−

= − − =

∫

E) 22 12

12

100812 36 212 5xV dxπ π

−

= − − =

∫

11. Set up and evaluate the integral that gives the volume of the solid formed by revolving

the region about the y-axis.

7 , 128 in the first quadranty x y= = A) 2 128

7 0

35849

V y dyπ π= =∫ D) 2 7

7 0

17929

V y dyπ π= =∫

B) 2 1287

0

17929

V y dyπ π= =∫ E) 1 7

7 0

35849

V y dyπ π= =∫

C) 1 1287

0

17929

V y dyπ π= =∫



12. Set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the y-axis.

1011 , 1, 0y x y x= = =

A) 11 15

0

532

V y dyπ π= =∫ D) 10 1

11 0

532

V y dyπ π= =∫

B) 11 110

0

516

V y dyπ π= =∫ E) 5 1

11 0

516

V y dyπ π= =∫

C) 11 15

0

516

V y dyπ π= =∫

13. Find the volume of the solid generated by revolving the region bounded by the graphs of

the equations about the given lines.

2 2, 20y x y x x= = − (i) x-axis; (ii) the line y = 102 A)

(i) 1003π ; 1400

3π

D)(i) 100000

3π ; 104000

3π

B) (i) 1000

3π ; 1400

3π

E) (i) 100

3π ; 104000

3π

C) (i) 1000

3π ; 104000

3π


the equations about the given lines.

2 2, 26x y x y y= = − (i) y-axis; (ii) the line x = 171 A)

(i) 21973

π ; 28733

π D)

(i) 1693π ; 2873

3π

B) (i) 371293

3π ; 380081

3π

E) (i) 169

3π ; 380081

3π

C) (i) 2197

3π ; 380081

3π



15. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

222 10 , 22y x x y x= − − = + (i) x-axis; (ii) the line y = 11 A)

(i) 11712815

π ; 19033315

π D)

(i) 19033330

π ; 5856415

π

B) (i) 380666

15π ; 234256

15π

E) (i) 58564

15π ; 190333

30π

C) (i) 190333

15π ; 117128

15π


the equations about the line y = 12.

, 11, 0y x y x= = = A) 1694

3π B) 847

3π C) 2057

3π D) π E) 2057

6π



21 , 3, 03

y x y x= = =

A) 72

5π B) 36

5π C) 18

5π D) 9

5π E) None of the above



sin , 0, 02

y x y x π= = ≤ ≤

A) 4

2ππ −

B) 4

4ππ −

C) 8

4ππ −

D) 2

4ππ −

E) 2

2ππ −



19. Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis.

1 , 0, 0, 811

y y xx

= = =+

x =

A) 27

11π B) 27ln

11π

C) 27ln22

π

D) 1911

ln π

E) 2722

π


the equations about the x-axis.

1 , 0, 5, 9y y x xx

= = = =

A) 4

45π B) 14

45π C) 2

45π D) 7

45π E) 19

90π


the equations about the x-axis. Verify your results using the integration capabilities of a graphing utility.

sin( ), 0, 0, 6

y x y x x π= = = =

A) 21 3 +

6 8π π

D)21 3 –

12 8π π

B) 21 3 +

12 8π π

E) 21 3 –

6 8π π

C) 21 3 –

12 4π π



22. Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.

216 , 0, 64y x x x y= − = = A) ( )( ) 8 2

0

40962 256 163

V x x x dxπ π= − − =∫

B) ( )( ) 8 2

0

20482 128 163

V x x x dxπ π= − − =∫

C) ( )( ) 8 2

0

40962 64 163

V x x x dxπ π= − − =∫

D) ( )( ) 8 2

0

20482 256 163

V x x x dxπ π= − − =∫

E) ( )( ) 8 2

0

20482 64 163

V x x x dxπ π= − − =∫

23. Use the shell method to set up and evaluate an integral that gives the volume of the solid

generated by revolving the plane region about the y-axis.

249 , 0y x y= − = A) 7 2

04 (49 ) 2401V x x dxπ π= − =∫ D) 7 2

–74 (49 ) 2401V x x dxπ π= − =∫

B) 7 2

0

24012 (49 )2

V x x dxπ π= − =∫ E) 0 2

–72 (49 ) 2401V x x dxπ π= − =∫

C) 7 2

–7

24012 (49 )2

V x x dxπ π= − =∫

24. Use the shell method to set up and evaluate the integral that gives the volume of the

solid generated by revolving the plane region about the x-axis.

8 , 0, y x y x= − = = 0 A) 8

0

5122 ( )(8 )3

V y y dyπ π= − =∫ D) 8

0

5122 ( )(8 )3

V y y dyπ π= + =∫

B) 8

0

2562 ( )(8 )3

V y y dyπ π= − =∫ E) 8

0

20482 ( )(8 )3


C) 8

0

20482 ( )(8 )3

V y y dyπ π= − + =∫



25. Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis.

, 0, 14y x y x= = = A) 7

0

96042 ( )(7 )3

V y y dyπ π= + =∫ D) 14

0

27442 ( )(7 )3


B) 7

0

13722 ( )(14 )3

V y y dyπ π= + =∫ E) 14

0

27442 ( )(14 )3

V y y dyπ π= − =∫

C) 7

0

96042 ( )(14 )3

V y y dyπ π= − =∫

26. Use the shell method to set up and evaluate the integral that gives the volume of the

solid generated by revolving the plane region about the x-axis.

7 , 0, 128y x x y= = = A) 1 128

7 0

17922 ( )15

V y y dyπ π= =∫ D) 1 128

7 0

2293762 ( )15

V y y dyπ π−

= =∫

B) 1 1287

0

4587522 ( )15

V y y dyπ π= =∫ E) 1 128

7 0

2293762 ( )15

V y y dyπ π= =∫

C) 1 1287

0

17922 ( )15

V y y dyπ π−

= =∫

27. Use the shell method to find the volume of the solid generated by revolving the plane

region about the line . 384x =

8 , 0, 256y x y x= = = A) 11534336

51V π=

D) 288358451

V π=

B) 576716851

V π= E) 1441792

51V π=

C) 1730150651

V π=



28. Use the disk or shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given line.

3, y 0, 9y x x= = = (i) the x-axis; (ii) the y-axis; (iii) the line x = 18 A)

(i) 95659387

π ; (ii) 1180985

π ; (iii) 1771475

π

B) (i) 4782969

7π ; (ii) 59049

5π ; (iii) 177147

5π

C) (i) 9565938

7π ; (ii) 118098

5π ; (iii) 118098

5π

D) (i) 4782969

7π ; (ii) 118098

5π ; (iii) 177147

5π

E) (i) 9565938

7π ; (ii) 59049

5π ; (iii) 118098

5π

29. Use the disk or shell method to find the volume of the solid generated by revolving the

region bounded by the graphs of the equations about the given line.

2

14 , 0, 1, 7y y x xx

= = = =

(i) the x-axis; (ii) the y-axis; (iii) the line y = 14 A)

(i) 2287

π ; (ii) 14ln(7)π ; (iii) 9487

π

B) (i) 456

7π ; (ii) 28ln(7)π ; (iii) 1896

7π

C) (i) 456

7π ; (ii) 7 ln(7)π ; (iii) 1896

7π

D) (i) 228

7π ; (ii) 7 ln(7)π ; (iii) 948

7π

E) (i) 228

7π ; (ii) 28ln(7)π ; (iii) 948

7π



30. Use the disk or shell method to find the volume of the solid generated by revolving the region bounded by the graph of the equation about the given line.

2 23 3 4x y+ =

23

(i) the x-axis; (ii) the y-axis A)

(i) 512105

π ; (ii) 2048105

π D)

(i) 2048105

π ; (ii) 512105

π

B) (i) 512

25π ; (ii) 512

25π

E) (i) 128

25π ; (ii) 2048

105π

C) (i) 2048

105π ; (ii) 2048

105π

31.

Find the arc length of the graph of the function 322 5

3y x= + over the interval [12,14].

A) ( )3 15 15 13 132

+ D) ( )2 15 15 13 13

3+

B) ( )3 15 15 13 132

− E) ( )2 15 15 13 13

3−

C) 15 15 13 13−

32. Find the arc length of the graph of the function

324y x 5= + over the interval [0,5].

A) 181 181 1108

+ D) 181 181 1

54−

B) 181 181 154

+ E) 181 181 1

108−

C) 181 181 18054

−

33.

Find the arc length of the graph of the function 233 5

2y x= + over the interval [1,1000].

A) 101 101 5 5− D) 3 3103 103 5 5− B) 33101 101 2 2− E) 103 103 2 2− C) 101 101 2 2−



34. Find the arc length of the graph of the function

3

1

16 2xy

x= + over the interval [1,2].

A) 15

4 B) 49

24 C) 49

12 D) 15 E) None of the above

8

35. Find the arc length of the graph of the function ( )

32 21 2

3x y= + over the interval

. 0 5y≤ ≤ A) 140

3 B) 139

6 C) 139

3 D) 137 E)

6472

36.

Find the arc length of the graph of the function 1 ( 3)3

x y= − y over the interval

. 1 144y≤ ≤ A) 880

3 B) 1760

3 C) 1768

3 D) 884

3 E) 1760

9

37. Find the area of the surface generated by revolving the curve about the x-axis.

31 , 0 15

15y x x= ≤ ≤

A) 325 2026 1

17π

−

D) 322026 1 π

+

B) 325 2026 1

9π

−

E) 325 136 1

17π

+

C) 325 2026 1

9π

+



38. Find the area of the surface generated by revolving the curve about the x-axis.

4 , 5y x x= ≤ 7≤ A) 3 3

2 21 44 363

π

−

D) 3 3

2 21 32 243

π

−

B) 3 32 22 32 24

3π

−

E) None of the above.

C) 3 32 22 44 36

3π

−

39. Find the area of the surface generated by revolving the curve about the y-axis.

3 10, 1 1000y x x= + ≤ ≤

A) 3 32 290001 10

27π+

D) 3 32 290001 5

27π−

B) 3 32 290001 10

27π−

E) 3 32 2901 1027

π−

C) 3 32 291 527

π+


2100 , 0 10y x x= − ≤ ≤

A) 3240001 1

3π−

D) 32401 16

π−

B) 3240001 1

6π−

E) 3240001 1

12π−

C) 32401 1

12π−




281 , 0 8y x x= − ≤ ≤ A) ( )32 9 17 π− D) ( )16 9 17 π−

B) ( )16 8 17 π− E) ( )18 9 17 π−

C) ( )36 9 17 π−

42. Determine the work done by lifting a 100 pound bag of sugar 7 feet.

A) 1,700 ft lb B) 170 ft ⋅ lb C) 7,000 ft⋅ ⋅ lb D) 70 ft ⋅ lb E) 700 ft ⋅ lb

43. A force of 8 pounds compresses a 20-inch spring 4 inches. How much work is done in compressing the spring from a length of 13 inches to a length of 9 inches?

A) 72 ft lb B) 77 ft ⋅ lb C) 82 ft⋅ ⋅ lb D) 74 ft ⋅ lb E) 67 ft ⋅ lb

44. A force of 270 Newtons stretches a spring 50 centimeters. How much work is done in stretching the spring from 30 centimeters to 80 centimeters? A) 158.5 N ⋅m D) 150.5 N ⋅m B) 153.5 N ⋅m E) 143.5 N ⋅m C) 148.5 N ⋅m

45. Neglecting air resistance and the weight of the propellant, determine the work done in

propelling a 7-ton satellite to a height of (i) 250 miles above Earth (ii) 450 miles above Earth Assume that Earth has a radius of 4000 miles. A) (i) 988.24 mi ton; (ii) 1,698.88 mi⋅ ⋅ ton B) (i) 2,470.59 mi ton; (ii) 4,247.19 mi⋅ ⋅ ton C) (i) 1,317.65 mi ton; (ii) 2,265.17 mi⋅ ⋅ ton D) (i) 1,647.06 mi ton; (ii) 2,831.46 mi⋅ ⋅ ton E) None of the above

46. A cylindrical water tank 6 meters high with a radius of 2 meters is buried so that the top

of the tank is 1 meter below ground level. How much work is done in pumping a full tank of water up to ground level? (The water weighs 9800 newtons per cubic meter.) A) 752,640π N ⋅m D) 564,480π N ⋅m B) 1,411,200π N ⋅m E) 1,693,440π N ⋅m C) 940,800π N ⋅m



47. An open tank has the shape of a right circular cone. The tank is 6 feet across the top and 5 feet high. How much work is done in emptying the tank by pumping the water over the top edge? Note: The density of water is 62.4 lbs per cubic foot. A) 1,170π ft ⋅ lb D) 702π ft ⋅ lb B) 1,755π ft ⋅ lb E) 2,106π ft ⋅ lb C) 936π ft ⋅ lb

48. A 14-foot chain that weighs 6 pounds per cubic foot hanging from a winch 14 feet

above ground level. Find the work done by the winch in winding up the entire chain. A) 1,058.4 ft lb B) 882 ft⋅ ⋅ lb C) 470.4 ft ⋅ lb D) 352.8 ft ⋅ lb E) 588 ft ⋅ lb

49. A quantity of gas with an initial volume of 9 cubic feet and a pressure of 1000 pounds

per square foot expands to a volume of 10 cubic feet. A) 568.946785 ft ⋅ lb D) 948.244641 ft ⋅ lb B) 1,422.366961 ft ⋅ lb E) 1,706.840354 ft ⋅ lb C) 758.595713 ft ⋅ lb

50. Find the center of mass of the point masses lying on the x-axis.

1 2 3

1 2 3

3, 3, 74, –10, –6

m m mx x x

= = == = =

A) 61–

13x = B) 57–

16x = C) 57–

13x = D) 58–

13x = E) 60–

13x =


1 2 3 4

1 2 3 4

4, 5, 3, 9–6, 1, 5, –10

m m m mx x x x

= = = == = = =

A) 91–

24x = B) 94–

21x = C) 13–

3x = D) 92–

21x = E) 95–

21x =


1 2 3 4 5

1 2 3 4 5

6 2 3 54 –10 –6 8 –2

m m m m mx x x x x

= = = = == = = = =

9

A) 2

5x = B) 11

28x = C) 11

25x = D) 8

25x = E) 7

25x =



53. Find the center of mass of the given system of point masses.

( ) ( ) (7 7 3

( , ) –2, –4 –9, –2 –1,7i

i i

mx y )

A) 80 21– , –17 17

x y= = D) 78 19– , –

17 17x y= =

B) 77 9– , –20 10

x y= = E) 81 22– , –

17 17x y= =

C) 77 18– , –17 17

x y= =


( ) ( ) ( ) ( ) (3 1 5 5

, –6, –7 5, –5 9,0 –8,9i

i i

mx y )

A) 5 1– , 14 7

x y= =1

D) 3 3– , 7 2

x y= =

B) 5 2– , 17 17

x y= =2

E) 9 9– , 14 7

x y= =

C) 4 1– , 7 1

x y= =94

)


( ) ( ) ( ) ( ) ( ) (6 3 6 4 8

, –4,8 –10, –1 2,6 3,4 –1, –4i

i i

mx y

A) 38 65– , 27 27

x y= = D) 4 6– ,

3 2x y= =

77

B) 7 3– , 6 1

x y= =45

E) 13 64– ,

9 2x y= =

7

C) 35 68– , 27 27

x y= =



56. Find Mx, My, and ( ),x y for the laminas of uniform density ρ bounded by the graphs of the equations.

677 , 0,y x y x= = = 2

A) 183500819xM ρ= , 1835008

5yM ρ= , 416 416, 5 1

x y= =9

B) 183500819xM ρ= ,M ,0y =

832 416, 11 21

x y= =

C) 0xM = , 1835008

5yM ρ= , 832 416, 9 1

x y= =7

D) 91750419xM ρ= , 1835008

5yM ρ= , 416 416, 5 2

x y= =1

E) 22937619xM ρ= , 1835008

5yM ρ= , 832 416, 11 19

x y= =

57. Find Mx, My, and ( ),x y for the laminas of uniform density ρ bounded by the graphs of

the equations.

216 , 0x y x= − = A)

0xM = , 819215yM ρ= , 80,

5x y= =

B) 0xM = , ,0yM = 0, 0x y= = C)

0xM = , 819215yM ρ= , 32 , 0

5x y= =

D) 819215xM ρ= , ,0yM =

8 8, 5 5

x y= =

E) 819215xM ρ= , ,0yM =

32 8, 5 5

x y= =



58. Set up and evaluate integrals for finding the area and moments about the x- and y-axes for the region bounded by the graphs of the equations. (Assume 1ρ = .)

6 12, 0, 0 21y x y x= + = ≤ ≤ A) ( ) ( ) ( ) 21 21 212 2

0 0 0

16 12 1,575; 6 12 72,954; 6 12 21,1682x yA x dx M x dx M x x dx= + = = + = = + =∫ ∫ ∫

B) ( ) ( ) ( ) 21 21 21 22 2

0 0 0

16 12 1,449; 6 12 72,954; 6 12 21,1682x yA x dx M x x dx M x x dx= + = = + = = + =∫ ∫ ∫

C) ( ) ( ) ( ) 21 21 212 2

0 0 0

16 12 1,449, 6 12 72,954; 6 12 21,6092x yA x dx M x dx M x x dx= + = = + = = + =∫ ∫ ∫

D) ( ) ( ) ( ) 21 21 212 2

0 0 0

13 6 1,449; 6 12 72,996; 6 12 21,1682x yA x dx M x dx M x x dx= + = = + = = + =∫ ∫ ∫

E) None of the above

59. Set up and evaluate integrals for finding the area and moments about the x- and y-axes for the region bounded by the graphs of the equations. (Assume 1ρ = .)

2121 , 0y x y= − = A) ( ) 11 2

0

5,3242 121 ;3

A x dx= − =∫ 0xM = by symmetry;

( ) 11 22

–11

1 1,121yM x dx= −∫288,408

2 1= −

5

B) ( ) 11 2

–11

5,3241213

A x dx= − − =∫ ; 0xM = by symmetry;

( ) 11 22

–11

1 1,121yM x dx= − =∫288,408

2 1−

5

C) ( ) ( ) 11 11 22 2

–11 –11

5,324 1 1,288,408121 ; 121 ; 03 2 15x yA x dx M x dx M= − − = = − = =∫ ∫

by symmetry D) ( ) ( ) 11 11 22 2

0 –11

5,324 1 1,288,4082 121 ; 121 ; 03 2 15x yA x dx M x dx M= − = = − = −∫ ∫ =

E) Both B and D F) Both A and C

60. Find the volume of the solid generated by rotating the circle ( )2 210 49x y− + = about

the y-axis. A) V 2140π= B) V 2980π= C) V 980π= D) V 140π= E) V 2490π=

61. Find the volume of the solid generated by rotating the circle ( )22 3x y 9+ − = about the

x-axis. A) V 227π= B) V 218π= C) V 54π= D) V 18π= E) V 254π=



62. Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

A) 249.6 lb B) 2,340 lb C) 468 lb D) 1,404 lb E) 62.4 lb

63. Find the fluid force on the vertical side of the tank, where the dimensions are given in

feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

A) 832 lb B) 1,040 lb C) 208 lb D) 93.6 lb E) 10.4 lb

64. Find the fluid force on the vertical side of the tank, where the dimensions are given in

feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

A) 140.4 lb B) 561.6 lb C) 280.8 lb D) 1,123.2 lb E) 2,246.4 lb



65. The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form.

A) 140.7 lb B) 5,276.25 lb C) 1,055.25 lb D) 562.8 lb E) 3,165.75 lb

66. The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds

per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form.

A) 562.8 lb B) 3,376.8 lb C) 2,251.2 lb D) 234.5 lb E) 46.9 lb

67. A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find

the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is 9 feet and the gasoline weighs 42 pounds per cubic foot.

A) 2,551.5 lb B) 1,275.75 lb C) 5,103 lb D) 10,206 lb E) 20,412 lb



Answer Key

1. B Section: 7.1

2. E Section: 7.1

3. D Section: 7.1

4. A Section: 7.1

5. C Section: 7.1

6. B Section: 7.1

7. A Section: 7.1

8. E Section: 7.1

9. D Section: 7.1

10. B Section: 7.2

11. A Section: 7.2

12. C Section: 7.2

13. D Section: 7.2

14. B Section: 7.2

15. C Section: 7.2

16. A Section: 7.2

17. A Section: 7.2

18. B Section: 7.2

19. D Section: 7.2

20. A Section: 7.2

21. D Section: 7.2

22. E Section: 7.3



23. B Section: 7.3

24. A Section: 7.3

25. E Section: 7.3

26. B Section: 7.3

27. A Section: 7.3

28. D Section: 7.3

29. B Section: 7.3

30. C Section: 7.3

31. E Section: 7.4

32. D Section: 7.4

33. C Section: 7.4

34. E Section: 7.4

35. A Section: 7.4

36. B Section: 7.4

37. B Section: 7.4

38. C Section: 7.4

39. B Section: 7.4

40. D Section: 7.4

41. E Section: 7.4

42. E Section: 7.5

43. A Section: 7.5

44. C Section: 7.5

45. D Section: 7.5




46. C Section: 7.5

47. A Section: 7.5

48. A Section: 7.5

49. D Section: 7.5

50. E Section: 7.6

51. B Section: 7.6

52. D Section: 7.6

53. A Section: 7.6

54. C Section: 7.6

55. A Section: 7.6

56. A Section: 7.6

57. C Section: 7.6

58. A Section: 7.6

59. C Section: 7.6

60. B Section: 7.6

61. E Section: 7.6

62. D Section: 7.7

63. B Section: 7.7

64. D Section: 7.7

65. E Section: 7.7

66. C Section: 7.7

67. A Section: 7.7

chapter 7: applications of integration -...

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