i need help calculus .
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questions.pdf
Math 1242 Sample Test #3 Name:
1. Set up the integral to compute the arclength of the curve 2/32xy and 30 x .
a) 3
0
22/3 )2(1 dxx b) 3
0
341 dxx c) 3
0
22/1 )3(1 dxx d) 3
0
91 dxx e) none of these
2. Which method will you use to evaluate the integral in #1.
a) A substitution to get 3
0
duu b) A substitution to get 3
0 9
1 duu
c) A substitution to get 28
1 9
1 duu d) A substitution to get
3
0 12
1 duu e) none of these
3. Compute the integral in #1 to find the arclength of the curve.
a) )12828( 9
1 b) )12828(
9
1 c) )12828(
27
2 d) )133(
27
2 e) none of these
4. Approximate the integral found in #1 using Simpson’s rule with n = 6. a) 10.899 b)10.889 c) 10.901 d)10.809 e) none of these
5. A force of 40 N is required to maintain a spring stretched from its natural length of 10cm to a length of 15cm. How much work is done in stretching the spring from 25cm to 35cm? a) 16 b) 24 c) 1600 d) 2400 e) none of these
6. (Good for a free response question) A tank has the shape of an inverted circular cone with height of 15 ft and base radius of 5 ft. It is filled with water to a height of 11 ft. Find the work required to empty the tank by pumping all of the water to the top
of the tank. (the density of the water is 62.5lb/ft 3
) a) 20800 b) 23400 c) 23600 d) 24600 e) none of these
7. (Good for a free response question) A tank has the shape of an inverted square-base pyramid with height of 12m and base width of 3m. It is filled with water to the top. Find the work required to empty the tank by pumping all of the water out of a
spout 1m above the top of the tank. (the density of the water is 1000kg/m 3
and gravity g = 9.8) a)1411200 b) 1411300 c) 1511200 d) 1411100 e) none of these
8. A circular swimming pool with height of 6 m and a diameter of 4 m. It is filled with water to the top. Find the work required
to empty half the tank only by pumping the water out of the top of the tank. (the density of the water is 1000kg/m 3
and gravity g = 9.8) a) 176400 b) 186400 c) 705600 d) 1411100 e) none of these
9. Let R be the region bounded by the graph of xy ln , the x-axis, and the line x = 2. Set up the integral to find the volume of the solid generated when R is revolved about the x-axis using the slicing method.
a) 2
0
2)(ln dxx b) 2
1
2)(ln dxx c) 2ln
0
)2(2 dyey y d) 2ln
0
22 )2( dye y e) 2
1
ln2 xdxx
10. Do #9 using the method of cylindrical shells.
a) 2
0
2)(ln dxx b) 2
1
2)(ln dxx c) 2ln
0
)2(2 dyey y d) 2ln
0
22 )2( dye y e) 2
1
ln2 xdxx
11. Let R be the region defined in #9. Set up the integral to find the volume of the solid generated when R is revolved about the y-axis using the slicing method.
a) 2
0
2)(ln dxx b) 2
1
2)(ln dxx c) 2ln
0
)2(2 dyey y d) 2ln
0
22 )2( dye y e) 2
1
ln2 xdxx
12. Do #11 using the method of cylindrical shells.
a) 2
0
2)(ln dxx b) 2
1
2)(ln dxx c) 2ln
0
)2(2 dyey y d) 2ln
0
22 )2( dye y e) 2
1
ln2 xdxx
13. Let S be the region bounded by the graph of 2xy and xy 3 . Find the volume of the solid generated when S is
revolved about the x-axis. a) 13.5 b)32.4 c) 23.6 d) 41.4 e) none of these
14. Find the volume of the solid generated when S of #13 is revolved about the y-axis. a) 13.5 b)32.4 c) 23.6 d) 41.4 e) none of these
15. Find the volume of the solid generated when S of #13 is revolved about the line y = -1. a) 13.5 b)32.4 c) 23.6 d) 41.4 e) none of these
16. Find the volume of the solid generated when S of #13 is revolved about the line x = 3. a) 13.5 b)32.4 c) 23.6 d) 41.4 e) none of these
17. Find the area of the region bounded by the curves 152 2 yx and yx . a) 42.6667 b) 13.5 c) 22.7831 d) 34.6667 e) none of these
18. Find the area of the region bounded by the curves 2,2 xyxy , and the x-axis.
a) 1/2 b) 2/3 c) 3/4 d) 4/5 e) 5/6
19. A 1600-lb elevator is suspended by a 200-ft cable that weighs 10lb/ft. How much work is required to raise the elevator from the basement to the top of a building 200 ft high? a) 320,000 b) 200,000 c) 520,000 d) 420,000 e) none of these
20. A force of 40 lb is required to maintain a spring stretched 7 in beyond its natural length. How much work is done in stretching the spring from its natural length of 15in to a length of 20in? a) 5.95 b) 8.32 c) 25.96 d) 71.43 e) none of these
21. (Good for a free response question) A hemispherical tank with radius of 7ft. is filled with water to the top. Find the work
required to empty the tank by pumping all of the water out of the side of the tank. (the density of the water is 62.5lb/ft 3
) a) 37420 b) 37520 c) 47520 d) 37540 e) none of these
22. Let R be the region in the first quadrant bounded by the graph of 4xy , x = 0, and the line y = 1. Set up the integral to find
the volume of the solid generated when R is revolved about the line x = -1 using the slicing method.
a) 1
0
4/12/1 )2( dyyy b) 1
0
2 )2( dxxx c) 1
0
4 )1(2 dxxx
d) 1
0
4/12/1 )2( dyyy e) 2
1
4/1 )1(2 dyyy
23. Do #22 using the method of cylindrical shells.
a) 1
0
4/12/1 )2( dyyy b) 1
0
2 )2( dxxx c) 1
0
4 )1(2 dxxx
d) 1
0
4/12/1 )2( dyyy e) 1
0
4 )1)(1(2 dxxx
24. Let R be the region defined in #22. Set up the integral to find the volume of the solid generated when R is revolved about the line y =2 using the slicing method.
a) 1
0
4/12/1 )2( dyyy b) 1
0
2 )2( dxxx c) 1
0
4 )1(2 dxxx
d) 1
0
84 )43( dxxx e) 2
1
4/1 )1(2 dyyy
25. Find the area of the region bounded by the curves xxy 42 and 22 xxy . a) 7.5 b) 9.0 c) 10.3 d) 6.8 e) none of these
26. Find the coordinates ( yx , ) of the centroid of the region S defined in #13 that is bounded by the graph of 2xy and xy 3 .
a) (1.5,4.5) b) (6.75, 32.4) c) (1.5, 3.6) d) (2.1, 5.1) e) none of these
27. Use your calculator to find the coordinates ( yx , ) of the centroid of the region R defined in #9 that is bounded by the graph of xy ln , the x-axis, and the line x = 2. a) (1.5, 0.5) b) (2, 0.693) c) (0.6363, 0.094) d) (1.647, 0.2438) e) cannot be found
28. Consider a lamina bounded by the curves 2xy , x = 1 and the x-axis. Find its area.
a) 1/3 b) 2/3 c) 3/4 d) 4/5 e) 5/6
29. If the density of the lamina in #28 is 4 , find the moment xM . a) 0.2 b) 0.3 c) 0.4 d) 0.5 e) 0.6
30. If the density of the lamina in #28 is 4 , find the moment yM . a) 0.5 b) 0.7 c) 1.0 d) 1.2 e) 1.5
31. If the density of the lamina in #28 is 4 , find the mass m of this lamina. a) 4/3 b) 2/3 c) 3/4 d) 4/5 e) 5/6
32. Find the coordinates ( yx , ) of the centroid of the lamina defined in #28. a) (1/3, 2/3) b) (1/2, 1/2) c) (3/4, 3/4) d) (3/4, 3/10) e) (0.5, 0.3)
33. The base of a solid is enclosed by the ellipse 100254 22 yx . Cross-sections perpendicular to the x-axis are isosceles right triangles with hypotenuse at the base. Calculate the volume of this solid. a) 64/3 b) 80/3 c) 40/3 d) 32/3 e) none of these.
34. Find the length of the curve 2
4
8
1
4 x
x y and 31 x . Show all your work.
