Name: Instructor: Multiple Choice 1.(5 Pts.) Which Integral Below Gives The

Name:

Instructor:

Multiple Choice

1.(5 pts.) Which integral below gives the√ volume of the solid of revolution obtained by rotating the bounded region between y = x and y = x2 about the line x = 3 ?

Z 1 √ Z 1 √ √ (a) 2 (x − 3)(x − x) dx (b) 2π ( x + x2)( x − x2) dx 0 0

Z 1 √ Z 1 3 √ (c) 2π (3 − x)( x − x2) dx (d) 2π x − ( x − x2) dx 0 0 2π Z 1 √ √ (e) 2π (3 − x)( x − x2) dx 0

2.(5 pts.) Find the volume of the solid of revolution obtained by rotating the region in the xy-plane bounded by y = x3 + 1, x = 1 and y = 1 about the y-axis. 11π 4π 3π 2π 2π (a) (b) (c) (d) (e) 3 13 7 5 9

2 Name:

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3.(5 pts.) Which integral below gives the arc length of the curve y = tan x between x = 0 π and x = ? 4 π π Z 4 p Z 4 p (a) 1 − sec4 x dx (b) 1 + sec4 x dx 0 0

π Z 1 rπ Z 4 p (c) + sec4 x dx (d) 1 + tan2 x dx 0 4 0

π Z 4 p (e) 1 + sec2 x tan2 x dx 0

4.(5 pts.) If the curve y = ln x between x = 1 and x = e is rotated about the y axis, which integral below gives the surface area?

Z e ln x Z e r 1 (a) 2π √ dx (b) 2π x 1 + dx 2 2 1 1 + x 1 x

Z e p Z e p (c) 2π x 1 + x2 dx (d) 2π ln x 1 + x2 dx 1 1

Z e r 1 (e) 2π x 1 − 2 dx 1 x

3 Name:

Instructor:

5.(5 pts.) A force of 6 lbs. compresses a spring a distance of 3 ft. How much work is done in compressing this spring 1 ft. from its original length?

(a) 3 ft.-lbs. (b) 4 ft.-lbs. (c) 5 ft.-lbs. (d) 1 ft.-lb. (e) 2 ft.-lbs.

6.(5 pts.) A bucket full of water is to be drawn from a well. The full bucket weighs 20 lbs., while the rope attached to the bucket weighs 0.2 lbs./ft. What is the work done (in foot pounds) in raising the bucket 10 ft. to just clear the top of the well. Remark: You will do some work in hauling up the 10 ft. of rope as well as in hauling up the bucket.

(a) 230 (b) 240 (c) 210 (d) 220 (e) 200

4 Name:

Instructor:

7.(5 pts.) If four point–masses, each of mass 1 unit, are placed at the points (1, 0), (0, 1), (1, −2) and (x, y) in the plane what value of x and value of y will cause the center of mass of the four-mass system to be the origin?

(a) (2, −1) (b) (0.5, 0.5) (c) (0, 0) (d) (−3, 2) (e) (−2, 1)

√ 9π 8.(5 pts.) The plane lamina bounded by y = 9 − x2 and y = 0 has area . Find the y 2 coordinate of its centroid. 6 4 5 11 (a) 0 (b) (c) (d) (e) π π π π

5 Name:

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9.(5 pts.) 4

2 B

1 A

-3 -2 -1 1 2 3

-1

Consider the region above. The coordinates of the center of mass of region A are (−2, 1) and the coordinates of the center of mass of region B are (1, 2). Region A has a mass of 6 units and region B has a mass of 3 units. Which point below is the center of mass of the two regions together?

4 2 3 π π (a) −1, (b) − , (c) − , (d) (0, 1) (e) (−1, −1) 3 3 2 2 2

10.(5 pts.) A cubical tank with side length 3 ft. is ﬁlled with a liquid. Find the ﬂuid force exerted by the liquid on any one vertical side if the liquid weighs 60 lbs./ft.3

(a) 830 lbs. (b) 820 lbs. (c) 800 lbs. (d) 790 lbs. (e) 810 lbs.

6 Name:

Instructor: Z 11.(5 pts.) Evaluate the integral 8x3 ln x dx.

1 (a) 2x4 ln x − x4 + C (b) 12x3(ln x)2 − 4x3 ln x + C 2 8 (c) 8x4 ln x − x3 + C (d) 2x4 ln x + 4x3(ln x)2 + C 3 (e) 8x3 ln x − 2x4 + C

1 Z 3 12.(5 pts.) Evaluate the deﬁnite integral x2 ex +1 dx. −1

e2 − 1 e2 + 7 (a) (b) (c) e2 − 5 (d) 3e2 − ln(e2) 3 3 e2 + e−2 (e) 6

7 Name:

Instructor: Z 13.(5 pts.) Evaluate the integral x sec2 x dx.

x2 (a) (sec2 x − tan2 x) (b) x tan x + ln | cos x| + C 2 1 1 1 1 (c) x2 sec2 x − x3 tan2 x (d) x2 sec2 x + x sec3 x + C 2 6 2 3 sec3 x (e) + C 3x

Z 14.(5 pts.) Evaluate the integral cos3 x sin2 x dx.

2 1 1 (a) (cos3 x sin3 x) + C (b) cos4 x sin2 x − cos3 x sin3 x + C 3 4 3 1 1 1 1 (c) cos4 x sin x − cos5 x + C (d) sin3 x − sin5 x + C 4 20 3 5 1 1 (e) cos4 x − cos6 x + C 4 6

8 Name:

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Partial Credit You must show your work on the partial credit problems to receive credit!

15.(10 pts.) Determine√ the surface area of the surface of revolution obtained by revolving the curve y = 2 x2 from x = 0 to x = 1 around the y–axis. (a) (5pts.) What is ds in terms of x and dx? (b) (3pts.) Write an integral for the requested surface area. (c) (2pts.) Do the integral.

9 Name:

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16.(10 pts.) A circular plate of radius 1 foot is in a vertical dam wall. Set up an integral to determine the ﬂuid force on the plate when the water rises to a level 10 ft above the centre of the plate. (Water weighs 64 lbs. per cubic ft.)

10 Name:

Instructor:

17.(10 pts.) Evaluate the integral: Z arctan (2x) dx.

11 Name: ANSWERS

Instructor: ANSWERS

Math 120 Exam II March 18, 2004

• The Honor Code is in eﬀect for this examination. All work is to be your own. • No calculators. • The exam lasts for 75 minutes. • Be sure that your name is on every page in case pages become detached. • Be sure that you have all 11 pages of the test. • The backs of pages may be used if you need additional room to work on a problem. Good Luck!

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (•) (d) (e) 9. (•) (b) (c) (d) (e) 2. (a) (b) (c) (•) (e) 10. (a) (b) (c) (d) (•) 3. (a) (•) (c) (d) (e) 11. (•) (b) (c) (d) (e) 4. (a) (•) (c) (d) (e) 12. (•) (b) (c) (d) (e) 5. (a) (b) (c) (•) (e) 13. (a) (•) (c) (d) (e) 6. (a) (b) (•) (d) (e) 14. (a) (b) (c) (•) (e) 7. (a) (b) (c) (d) (•) 8. (a) (b) (•) (d) (e)

DO NOT WRITE IN THIS BOX!

Total multiple choice:

15.

16.

17.

Total: Name:

Instructor:

Math 120 Exam II March 18, 2004

PLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 9. (a) (b) (c) (d) (e) 2. (a) (b) (c) (d) (e) 10. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 11. (a) (b) (c) (d) (e) 4. (a) (b) (c) (d) (e) 12. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 13. (a) (b) (c) (d) (e) 6. (a) (b) (c) (d) (e) 14. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e) 8. (a) (b) (c) (d) (e)

DO NOT WRITE IN THIS BOX!

Total multiple choice:

15.

16.

17.

Total: