1.

 A. 0 B. 1/4 C. 1/2 D. 1 E. ¥
solution

2. What is the average value of xe^x2 on the interval [2, 4]?

 A. (e16 - e4)/4 B. (e16 - e4)/2 C. (2e16 - e4)/2 D. 2e16 - e4 E. 2e16 + e4
solution

3. Let F(x) be a strictly decreasing continuously differentiable function on [a, b]. Then must equal to

 A. |F(b)| - |F(a)| B. F(a) - F(b) C. F(b)-F(a) D. |F(a)| - |F(b)| E. F(-b) - F(-a)
solution

4. Let f(x) = 2x + 1 for 0 £ x £ 1. If the interval [0,1] is partitioned into 4 subintervals of equal length, then what is the smallest Riemann sum for f(x) and this partition?

 A. 7/4 B. 15/8 C. 2 D. 7/2 E. 7
solution

5.

 A. -p 2/2 B. -p C. 0 D. p E. p 2
solution

6. What is the area of the closed region bounded by x = -1, x = 0, y = x2, and y = x3 ?

 A. 1/12 B. 1/6 C. 1/4 D. 5/12 E. 7/12
solution

7. The closed region in the first quadrant bounded by the curves y = x3 and y = x(1/3) is rotated about the x-axis. What is the volume of the resulting solid?

 A. 1/2 B. 128p /455 C. 16p /35 D. p /2 E. 32p /35
solution

8. What is the x-coordinate of the centroid of the closed region

R = {(x,y)Î R2: 0 £ x £ 1, (1-x2)1/2 £ y £ 1} ?

 A. 1/6 B. 1/3 C. 2/(3(4-p )) D. 4/3p E. 6/(4-p )
solution

9.

 A. -2 B. -1 C. 1 D. 2 E. does not exist
solution

10.

 A. 5/6 - 3 ln 6 B. 61/6 + 3 ln 6 C. 5/6 + 3 ln 6 D. 17/6 + 3 ln 6 E. 17/6 - 3 ln 6
solution

11.

 A. -2 + (9/4)ln 3 B. -4 + (9/2)ln 3 C. -(1/4) + (9/4)ln 3 D. -(5/2) + (9/2)ln 3 E. -2 + (9/2)ln 3
solution

12. What is the area of the closed region bounded by y = x2 - |x| and the x-axis, between x = -1 and x = 1?

 A. 1/12 B. 1/6 C. 1/3 D. 2/3 E. 5/6
solution

13. What is the average value of the function f(x) = x sin(x2) over the interval [2,4]?

 A. sin 4 + 2 sin 16 B. -sin 4 + 2 sin 16 C. (cos 4 - cos 16)/2 D. (cos 4 - cos 16)/4 E. (-cos 4 + cos 16)/4

solution

14.

 A. (1/12)ln 8 B. 3/8 C. û(5/12) D. ln 8 E. does not exist
solution

15.

 A. ln (1 + e-1) B. - ln (1 + e-1) C. ln (1 + e) D. arctan (e1/2) E. does not exist
solution

16.

 A. 3/8 B. 2/3 C. 3/2 D. 9/4 E. 8/3
solution

17. If and then

 A. -3 B. -1 C. 0 D. 1 E. 3
solution

18. Let

.

Then

 A. 0 B. 1/3 C. 2/3 D. 1 E. Does not exist.
solution

19.

 A. (p 3/256)cos(p /8) B. (Ö 2/32)(p 2+8p -32) C. (p 2/32)Ö 2 D. (Ö 2/32)(p 2-8p +32) E. (p 3/384)(2-Ö 2)
solution

20. Let S be the closed region in the first quadrant of the xy-plane bounded by y = 6x2, y = 0, x = 0, and x = 1. What is the volume of the solid obtained by revolving S about the line x = -1?

 A. 3p B. 7p C. 36p /5 D. 8p E. 56p /5
solution

21. The region S is bounded by y = x2 - 2x + 3, y = 0, x = 0, and x = 9. Which of the following is the approximation to the area of S obtained by computing the sum of the areas of the 3 inscribed rectangles with bases [0,3], [3,6], and [6,9] (lower Riemann sum)?

 A. 105 B. 108 C. 117 D. 189 E. 297
solution

22.

 A.ln (x2+2x+3)+C B.(1/(2x+2))ln(x2+2x+3)+C C.-(x+1)/(x2+2x+3)2 +C D.Ö 2arctan(x+1)+C E.(1/Ö 2)arctan((x+1)/Ö 2)+C

solution

23. Let f(x) = x2. For what value of x does f(x) equal the average value of f on the closed interval [2,5]?

 A. Ö (107/9) B. Ö 13 C. Ö (125/9) D. Ö (133/9) E. 7
solution

24. If and , then

 A. -31 B. -19 C. 11 D. 30 E. 49
solution

25.

 A. -(47/576) B. 1/6 C. ln (8/9) D. ln 2 E. ln (32/9)
solution

26. Let S be the closed region in the first quadrant of the xy-plane bounded by y = sin(p x/2) and y = x for 0 £ x £ 1. What is the volume of the closed region in R3 obtained by revolving S about the x-axis?

 A. 2 - (p /2) B. p /6 C. p /3 D. p /2 E. (2p )/3
solution

27. Let f(x) = x2(x3 + 1)4. What is the area under the graph of f over the closed interval [-1,1]?

 A. 0 B. 32/15 C. 62/15 D. 64/15 E. 32/5
solution

28. What is the average value of y = (2x+1)1/2 over the interval [4,12]?

 A. Ö 3 - 1/3 B. 2(Ö 3 - 1/3) C. 4 D. 49/12 E. Ö 17
solution

29.

 A. 2 - ln(2) B. ln(2) - 1/2 C.2 tan2(2) + 4 ln[cos(2)] D. 2 ln[sec(2)]-sin2(2) E. 2 - p /2
solution

30.

 A. 2/p B. 25/8 C. 2 D. (1+2p )/p 2 E. 2p
solution