Calculus AB and Calculus BC

CHAPTER 7 Applications of Integration to Geometry

Practice Exercises

Part A. Directions: Answer these questions without using your calculator.

AREA

In Questions 1–11, choose the alternative that gives the area of the region whose boundaries are given.

1. The curve of y = x2, y = 0, x = −1, and x = 2.

(A) Image

(B) Image

(C) 3

(D) 5

(E) none of these

2. The parabola y = x2 − 3 and the line y = 1.

(A) Image

(B) 32

(C) Image

(D) Image

(E) none of these

3. The curve of x = y2 − 1 and the y-axis.

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

4. The parabola y2 = x and the line x + y = 2.

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

5. The curve of Image the x-axis, and the vertical lines x = −2 and x = 2.

(A) Image

(B) Image

(C)

(D) π

(E) none of these

6. The parabolas x = y2 − 5y and x = 3yy2.

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

7. The curve of Image and x + y = 3.

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

8. In the first quadrant, bounded below by the x-axis and above by the curves of y = sin x and y = cos x.

(A) Image

(B) Image

(C) 2

(D) Image

(E) Image

9. Bounded above by the curve y = sin x and below by y = cos x from Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

10. The curve y = cot x, the line Image and the x-axis.

(A) ln 2

(B) Image

(C) 1

(D) Image

(E) 2

11. The curve of y = x3 − 2x2 − 3x and the x-axis.

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

12. The total area bounded by the cubic x = y3y and the line x = 3y is equal to

(A) 4

(B) Image

(C) 8

(D) Image

(E) 16

13. The area bounded by y = ex, y = 2, and the y-axis is equal to

(A) 3 − e

(B) e2 − 1

(C) e2 + 1

(D) 2 ln 2 − 1

(E) 2 ln 2 − 3

14. The area enclosed by the ellipse with parametric equations x = 2 cos θ and y = 3 sin θ equals

(A)

(B) Image

(C)

(D) Image

(E) none of these

BC ONLY

15. The area enclosed by one arch of the cycloid with parametric equations x = θ − sin θ and y = 1 − cos θ equals

(A) Image

(B)

(C)

(D)

(E) none of these

BC ONLY

16. The area enclosed by the curve y2 = x(1 − x) is given by

(A) Image

(B) Image

(C) Image

(D) π

(E)

BC ONLY

17. The figure below shows part of the curve of y = x3 and a rectangle with two vertices at (0,0) and (c, 0). What is the ratio of the area of the rectangle to the shaded part of it above the cubic?

Image

(A) 3:4

(B) 5:4

(C) 4:3

(D) 3:1

(E) 2:1

VOLUME

In Questions 18–24 the region whose boundaries are given is rotated about the line indicated. Choose the alternative that gives the volume of the solid generated.

18. y = x2, x = 2, and y = 0; about the x-axis.

(A) Image

(B)

(C) Image

(D) Image

(E) Image

19. y = x2, x = 2, and y = 0; about the y-axis.

(A) Image

(B)

(C) Image

(D)

(E) Image

20. The first quadrant region bounded by y = x2, the y-axis, and y = 4; about the y-axis.

(A)

(B)

(C) Image

(D) Image

(E) Image

21. y = x2 and y = 4; about the x-axis.

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

22. y = x2 and y = 4; about the line y = 4.

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

23. An arch of y = sin x and the x-axis; about the x-axis.

(A) Image

(B) Image

(C) Image

(D) π2

(E) π(π − 1)

24. A trapezoid with vertices at (2,0), (2, 2), (4,0), and (4,4); about the x-axis.

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

25. The base of a solid is a circle of radius a, and every plane section perpendicular to a diameter is a square. The solid has volume

(A) Image

(B)a3

(C)a3

(D) Image

(E) Image

26. The base of a solid is the region bounded by the parabola x2 = 8y and the line y = 4, and each plane section perpendicular to the y-axis is an equilateral triangle. The volume of the solid is

(A) Image

(B) Image

(C) Image

(D) 32

(E) none of these

27. The base of a solid is the region bounded by y = e−x, the x-axis, the y-axis, and the line x = 1. Each cross section perpendicular to the x-axis is a square. The volume of the solid is

(A) Image

(B) e2 − 1

(C) Image

(D) Image

(E) Image

ARC LENGTH

28. The length of the arc of the curve y2 = x3 cut off by the line x = 4 is

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

BC ONLY

29. The length of the arc of y = ln cos x from Image equals

(A) Image

(B) 2

(C) Image

(D) Image

(E) Image

BC ONLY

IMPROPER INTEGRALS

30. Image

(A) 1

(B) Image

(C) −1

(D) Image

(E) none of these

31. Image

(A) 1

(B) Image

(C) Image

(D) −1

(E) none of these

32. Image

(A) Image

(B) Image

(C) 3

(D) 1

(E) none of these

33. Image

(A) 6

(B) Image

(C) Image

(D) 0

(E) none of these

34. Image

(A) 2

(B) −2

(C) 0

(D) Image

(E) none of these

35. Image

(A) −2

(B) Image

(C) 2

(D) Image

(E) none of these

BC ONLY

In Questions 36–40, choose the alternative that gives the area, if it exists, of the region described.

36. In the first quadrant under the curve of y = e−x.

(A) 1

(B) e

(C) Image

(D) 2

(E) none of these

37. In the first quadrant under the curve of y = xex2.

(A) 2

(B) Image

(C) Image

(D) Image

(E) none of these

38. In the first quadrant above y = 1, between the y-axis and the curve xy = 1.

(A) 1

(B) 2

(C) Image

(D) 4

(E) none of these

39. Between the curve Image and the x-axis.

(A)

(B)

(C)

(D) π

(E) none of these

40. Above the x-axis, between the curve Image and its asymptotes.

(A) Image

(B) π

(C)

(D)

(E) none of these

In Questions 41 and 42, choose the alternative that gives the volume, if it exists, of the solid generated.

41. Image at the left by x = 1, and below by y = 0; about the x-axis.

(A) Image

(B) π

(C)

(D)

(E) none of these

42. The first-quadrant region under y = e−x ; about the x-axis.

(A) Image

(B) π

(C)

(D)

(E) none of these

Part B. Directions: Some of the following questions require the use of a graphing calculator.

AREA

In Questions 43–47, choose the alternative that gives the area of the region whose boundaries are given.

43. The area bounded by the parabola y = 2 − x2 and the line y = x − 4 is given by

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

BC ONLY

Image

44. The area enclosed by the hypocycloid with parametric equations x = cos3 t and y = sin3 t as shown in the above diagram is

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

BC ONLY

45. Suppose the following is a table of ordinates for y = f (x), given that f is continuous on [1, 5]:

x

1

2

3

4

5

y

1.62

4.15

7.5

9.0

12.13

If a trapezoid sum in used, with n = 4, then the area under the curve, from x = 1 to x = 5, is equal, to two decimal places, to

(A) 6.88

(B) 13.76

(C) 20.30

(D) 25.73

(E) 27.53

46. The area A enclosed by the four-leaved rose r = cos 2θ equals, to three decimal places,

(A) 0.785

(B) 1.571

(C) 2.071

(D) 3.142

(E) 6.283

BC ONLY

47. The area bounded by the small loop of the limaçon r = 1 − 2 sin θ is given by the definite integral

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

VOLUME

In Questions 48–54 the region whose boundaries are given is rotated about the line indicated. Choose the alternative that gives the volume of the solid generated.

48. y = x2 and y = 4; about the line y = −1.

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

49. y = 3xx2 and y = 0; about the x-axis.

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

50. y = 3xx2 and y = x; about the x-axis.

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

51. y = ln x, y = 0, x = e; about the line x = e.

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

52. The curve with parametric equations x = tan θ, y = cos2 θ, and the lines x = 0, x = 1, and y = 0; about the x-axis.

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

53. A sphere of radius r is divided into two parts by a plane at distance h (0 < h < r) from the center. The volume of the smaller part equals

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

CHALLENGE

54. If the curves of f (x) and g(x) intersect for x = a and x = b and if f (x) > g(x) > 0 for all x on (a, b), then the volume obtained when the region bounded by the curves is rotated about the x-axis is equal to

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

ARC LENGTH

55. The length of one arch of the cycloid Image equals

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

56. The length of the arc of the parabola 4x = y2 cut off by the line x = 2 is given by the integral

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

BC ONLY

57. The length of x = et cos t, y = et sin t from t = 2 to t = 3 is equal to

(A) Image

(B) Image

(C) 2(e3e2)

(D) e3 (cos 3 + sin 3) − e2 (cos 2 + sin 2)

(E) none of these

CHALLENGE

IMPROPER INTEGRALS

58. Which one of the following is an improper integral?

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

59. Which one of the following improper integrals diverges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

60. Which one of the following improper integrals diverges?

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image