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. (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