Calculus AB and Calculus BC

CHAPTER 6 Definite Integrals

Practice Exercises

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

1. Image

(A) Image

(B) 0

(C) Image

(D) −2

(E) −1

2. Image

(A) Image

(B) Image

(C) 1 − ln 2

(D) Image

(E) 1

3. Image

(A) 1

(B) −2

(C) 4

(D) −1

(E) 2

4. Image

(A) 2

(B) Image

(C) Image

(D) 6

(E) Image

5. Image

(A) ln 3

(B) Image

(C) Image

(D) ln Image

(E) Image

6. Image

(A) 1

(B) Image

(C) Image

(D) −1

(E) 2

7. Image

(A) Image

(B) 6

(C) Image

(D) 0

(E) 4

8. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

9. Image

(A) Image

(B) 0

(C) Image

(D) Image

(E) Image

10. Image

(A) Image

(B) 1 − e

(C) Image

(D) Image

(E) Image

11. Image

(A) e − 1

(B) Image

(C) 2(e − 1)

(D) Image

(E) Image

12. Image

(A) 2

(B) Image

(C) −1

(D) Image

(E) −2

13. Image

(A) −ln 2

(B) Image

(C) Image

(D) Image

(E) ln 2

14. If we let x = 2 sin θ, then Image is equivalent to

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

15. Image

(A) Image

(B) Image

(C) 1

(D) Image

(E) 0

16. Image

(A) Image

(B) Image

(C) 0

(D) 1

(E) e − 1

17. Image

(A) −1

(B) e + 1

(C) 1

(D) e − 1

(E) Image

BC ONLY

18. Image

(A) ln 2

(B) Image

(C) Image

(D) Image

(E) Image

19. Image

(A) Image

(B) Image

(C) Image

(D) ln 3

(E) Image

20. Image

(A) Image

(B) Image

(C) Image

(D) −1

(E) Image

21. Image

(A) Image

(B) 1

(C) Image

(D) Image

(E) −1

22. Image

(A) e

(B) 2 + e

(C) Image

(D) 1 + e

(E) e − 1

23. Image

(A) ln 2

(B) e

(C) 1 + e

(D) −ln 2

(E) Image

24. If we let x = tan θ, then Image is equivalent to

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

25. If the substitution Image is used, then Image is equivalent to

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

26. The table above shows some values of continuous function f and its first derivative. Evaluate Image

x

f (x)

f (x)

0

11

3

2

15

2

4

16

−1

6

12

−3

8

7

0

(A) −1/2

(B) −3/8

(C) 3

(D) 4

(E) none of these

27. Using M(3), we find that the approximate area of the shaded region below is

(A) 9

(B) 19

(C) 36

(D) 38

(E) 54

Image

28. The graph of a continuous function f passes through the points (4,2), (6,6), (7,5), and (10,8). Using trapezoids, we estimate that Image

(A) 25

(B) 30

(C) 32

(D) 33

(E) 41

29. The area of the shaded region in the figure is equal exactly to ln 3. If we approximate ln 3 using L(2) and R(2), which inequality follows?

Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

30. Let Image We estimate A using the L, R, and T approximations with n = 100 subintervals. Which is true?

(A) L < A < T < R

(B) L < T < A < R

(C) R < A < T < L

(D) R < T < A < L

(E) The order cannot be determined.

31. Image

(A) Image

(B) 4

(C) Image

(D) 5

(E) Image

32. Image

(A) Image

(B) Image

(C) 5

(D) Image

(E) Image

33. The average value of Image on its domain is

(A) 2

(B) 4

(C)

(D)

(E) none of these

34. The average value of cos x over the interval Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

35. The average value of csc2 x over the interval from Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

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

36. Find the average value of function f, as shown in the graph below, on the interval [0,5].

Image

(A) 2

(B) 4

(C) 5

(D) 7

(E) 8

37. The integral Image gives the area of

(A) a circle of radius 4

(B) a semicircle of radius 4

(C) a quadrant of a circle of radius 4

(D) an ellipse whose semimajor axis is 4

(E) none of these

38. Image

(A) 0.25

(B) 0.414

(C) 1.000

(D) 1.414

(E) 2.000

Use the graph of function f, shown below, for questions 39–42.

Image

39. In which of these intervals is there a value c for which f (c) is the average value of f over the interval [0,6]?

I. [0,2]

II. [2,4]

III. [4,6]

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) none of these, because f is not differentiable on [0,6]

40. Image

(A) −2

(B) Image

(C) 0

(D) Image

(E) 2

41. Let g(x) = Image then g (1)

(A) = 3.

(B) = 4.

(C) = 6.

(D) = 8.

(E) does not exist, because f is not differentiable at x = 2.

42. Let h(x) = x2f (x). Find Image

(A) 22

(B) 38

(C) 58

(D) 70

(E) 74

43. If f (x) is continuous on the closed interval [a,b], then there exists at least one number c, a < c < b, such that Image is equal to

(A) Image

(B) f (c)(b − a)

(C) f (c)(ba)

(D) Image

(E) f (c)[f (b) − f (a)]

44. If f (x) is continuous on the closed interval [a,b] and k is a constant, then Image is equal to

(A) k(ba)

(B) k[f (b) − f (a)]

(C) kF(ba), where Image

(D) Image

(E) Image

45. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) none of these

46. If Image then F (u) is equal to

(A) −6u(2 − u2)2

(B) Image

(C) (2 − u2)3 − 1

(D) (2 − u2)3

(E) −2u(2 − u2)3

47. Image

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

48. If x = 4 cos θ and y = 3 sin θ, then Image is equivalent to

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

49. A curve is defined by the parametric equations x = 2a tan θ and y = 2a cos2 θ, where 0 Image θ Image π. Then the definite integral Image is equivalent to

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

50. A curve is given parametrically by x = 1 − cos t and y = t − sin t, where 0 Image t Image π. Then Image is equivalent to

(A) Image

(B) Image

(C) Image

(D) Image

(E) Image

BC ONLY

51. When Image is estimated using n = 5 subintervals of equal width, which is (are) true?

I. Image

II. Image

III. Image

(A) II only

(B) III only

(C) I and II only

(D) I and III only

(E) II and III only

52. Find the value of x at which the function y = x2 reaches its average value on the interval [0,10].

(A) 4.642

(B) 5

(C) 5.313

(D) 5.774

(E) 7.071

53. The average value of Image on the interval 0 ≤ x ≤ 5 is

(A) 8

(B) 9.2

(C) 16

(D) 23

(E) undefined because f is not differentiable on this interval