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172Sp2001-copyright Joe Kahlig
Math 172
Sample Questions Exam I
1. Sketch the region bounded by the graphs of the equations and find its area. Find exact values.
(a) y = e−x , y = e3x , x = −1, and x = 2
(b) y 2 = 4 + x, y 2 + x = 2, y = 0, and y = 3
2. Find the number A such that the line x = A bisects the area under the curve y = 1/x2 , 1 ≤ x ≤ 4.
3. Find the volume of the solid generated by the region that is between y = 0 and y = 3 and is
also bounded by x = 0, and x = 4y − y 2 , if the region is rotated about x = 4.
4. Find the volume of the solid generated by the region bounded by y = x3 , x = 1, x = 2, and
y = 1, if the region is rotated about y = 1.
5. Find the volume, using cylindrical shells, of the solid bounded by y = −x2 + 4x − 2, x = 1,
x = 3, y = 0, and it is rotated about x = 4.
6. A force of 30N is required to maintain a spring stretched from its natrual length of 12cm to a
length of 15cm. How much work is done stretching the spring from 12cm to 20cm?
7. Find the average value of f (x) = sin2 (x) cos(x) on the interval [−π/2, π/4]
8. Compute these integrals. Show all work.
(a)
R
Z
(b)
Z
(c)
Z
(d)
sin−1 x dx
e2x cos x dx
sin5 (3x)dx
x sin2 xdx
Z
(e)
cos 6x cos 3xdx
Z
(f)
Z
(g)
tan5 x sec3 xdx
tan3 x sec4 xdx
Z
x
dx
+ 2x + 10
Z
2x + 3
√
(i)
dx
9 − 8x − x2
Z
3
√
(j)
4x2 − 25
Z 4
x + x3 + 8x2 + 16
(k)
dx
x(x2 + 4)2
(h)
x2