MAC 2312 Spring 13 Ref: 675181
Exam 1 Review
Mr. Guillen
Exam 1 will be on 02/01/13 and covers the following sections: 6.1, 6.2, 6.3, 6.4, 6.5, 6.6.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.
1) y = x, y = 0, x = 2, x = 4
1)
2) y =
x, y = 0, x = 0, x = 6
2)
3) y =
2x + 3, y = 0, x = 0, x = 1
3)
4) y =
5) y =
1
, y = 0, x = 1, x = 2
x
sin 4x, y = 0, 0 ≤ x ≤
4)
π
4
5)
6) y = 2x, y = 2, x = 0
6)
7) y = - 3x + 6, y = 3x, x = 0
7)
8) y = x2 , y = 36, x = 0
8)
9) y = x2 + 1, y = 3x + 1
9)
Find the volume of the solid generated by revolving the region about the y-axis.
y2
10) The region enclosed by x =
, x = 0, y = - 5, y = 5
5
11) The region enclosed by x = y1/3, x = 0, y = 27
12) The region enclosed by x =
13) The region enclosed by x =
6
, x = 0, y = 1, y = 4
y
sin 4y, 0 ≤ y ≤
π
,x=0
8
14) The region in the first quadrant bounded on the left by y = x3 , on the right by the line
x = 2, and below by the x-axis
10)
11)
12)
13)
14)
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
and lines about the y-axis.
x
15) y = 3x, y = - , x = 1
15)
3
16) y = 3x, y = 6x, x = 3
16)
17) y = 2x 2 , y = 2 x
17)
18) y = x2 , y = 4 + 3x, for x ≥ 0
18)
19) y = 50 - x2 , y = x2 , x = 0
19)
20) y =
5
, y = 0, x = 1, x = 25
x
20)
4
, y = 0, x = 2, x = 4
x
21)
22) y = x2 - 5, y = 4x, x = 0, for x ≥ 0
22)
21) y =
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
and lines about the x-axis.
23) x = 6 y, x = - 6y, y = 1
23)
24) x = 3y2 , x = - 3y, y = 3
24)
25) y =
25)
x, y = 0, y = x - 6
26) y = 5x, y = 10x, y = 5
26)
27) y = 8x 3 , y = 8x, for x ≥ 0
27)
28) x = 18 - y2 , x = y2 , y = 0
28)
Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves
about the given lines.
29) y = 4x,
y = x2 ; revolve about the y-axis
29)
30) y = 2x,
y = 0,
x = 2; revolve about the x-axis
30)
31) y = 5x,
y = 0,
x = 2; revolve about the line x = -3
31)
Find the length of the curve.
32) y = 3x 3/2 from x = 0 to x =
5
9
33) y = (16 - x2/3) 3/2 from x = 1 to x = 64
32)
33)
34) y =
1 3 1
x +
from x = 1 to x = 3
6
2x
34)
35) x =
y4
1
from y = 1 to y = 2
+
8
4y2
35)
36) x =
1 3/2
y
- y 1/2 from x = 9 to x = 16
3
36)
37) y =
3 4/3
(x
- 2x2/3) from x = 1 to x = 27
8
37)
38) x =
2
(y - 1) 3/2 from y = 1 to y = 4
3
38)
Find the area of the surface generated by revolving the curve about the indicated axis.
39) y = x3 /14, 0 ≤ x ≤ 4; x-axis
39)
40) x = y3 /14, 0 ≤ y ≤ 4; y-axis
40)
41) y =
41)
x, 3/2 ≤ x ≤ 9/2; x-axis
42) x = 3 4 - y, 0 ≤ y ≤ 15/4; y-axis
42)
Solve the problem.
43) The spring of a spring balance is 6.0 in. long when there is no weight on the balance, and it
is 8.9 in. long with 8.0 lb hung from the balance. How much work is done in stretching it
from 6.0 in. to a length of 14.8 in.?
43)
44) It took 1930 J of work to stretch a spring from its natural length of 2 m to a length of 4 m.
Find the springʹs force constant.
44)
45) A force of 3 N will stretch a rubber band 4 cm. Assuming Hookeʹs law applies, how much
work is done on the rubber band by a 9 N force?
45)
46) A bathroom scale is compressed
1
in. when a 190 lb person stands on it. Assuming that
4
46)
the scale behaves like a spring that obeys Hookeʹs law, how much does someone who
1
compresses the scale in. weigh?
8
47) A force of 1300 lb compresses a spring from its natural length of 17 in. to a length of 14 in.
How much work is done in compressing it from 14 in. to 7 in.?
Find the center of mass of a thin plate of constant density covering the given region.
48) The region bounded by y = x2 and y = 5
47)
48)
49) The region bounded by y = 10 - x and the axes
49)
50) The region bounded by y = x4 , x = 2, and the x-axis
50)
51) The region bounded by the parabola y = 49 - x2 and the x-axis
51)
52) The region enclosed by the parabolas y = - x2 + 18 and y = x2
52)
53) The region bounded by the parabola x = y2 and the line x = 9
53)
Find the center of mass of a thin plate covering the given region with the given density function.
54) The region bounded by the parabola y = 16 - x2 and the x-axis, with density δ(x) = 4x2
55) The region bounded below by the parabola y = x2 and above by the line y = x + 2, with
density δ(x) = 9x 2
56) The region between the x-axis and the curve y =
5
x2
, 1 ≤ x ≤ 2, with density δ(x) =
5
x2
54)
55)
56)
57) The region enclosed by the parabolas y = 8 - x2 and y = x2 , with density δ(x) = x2
57)
58) The region bounded by x = y2 and the line x = 4, with density δ(x) = y2
58)
59) The region between the curve y =
x
3
and the x-axis from x = 1 to x = 9, with density δ(x) =
x
59)
Answer Key
Testname: MAC_2312_SPRING_13_EXAM_1_REVIEW
1)
56
π
3
2) 18π
3) 4π
1
4) π
2
5)
1
π
2
6)
8
π
3
7) 18π
31104
8)
π
5
9)
207
π
5
10) 50π
729
11)
π
5
12) 27π
π
13)
4
14)
64
π
5
15)
20
π
9
16) 54π
3
17) π
5
18) 64π
19) 625π
2480
20)
π
3
21) 16π
875
22)
π
6
23)
44
π
5
24)
351
π
2
25)
63
π
2
26)
25
π
3
27)
256
π
21
28) 81π
Answer Key
Testname: MAC_2312_SPRING_13_EXAM_1_REVIEW
29)
128
π
3
30)
32
π
3
31)
260
π
3
32)
335
243
33) 90
14
34)
3
35)
33
16
36)
40
3
37) 36
14
38)
3
39)
1132
π
49
40)
1132
π
49
41)
19 19 7 7
π
6
6
42)
125
- 5 10 π
2
43)
44)
45)
46)
47)
110 lb·in.
965 N/m
0.54 J
95 lb
20,000 lb·in.
48) x = 0, y = 3
10
10
49) x =
,y=
3
3
50) x =
5
40
,y=
3
9
51) x = 0, y =
98
5
52) x = 0, y = 9
27
53) x =
,y=0
5
54) x = 0, y =
55) x =
32
7
8
118
,y=
7
49
Answer Key
Testname: MAC_2312_SPRING_13_EXAM_1_REVIEW
56) x =
3
5
,y=
2
4
57) x = 0, y = 4
20
58) x =
,y=0
7
59) x =
13
1
,y=
3
2
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