Review for Chapter 6
1. Let the base of a solid be a 30
o

60
o
right triangle, with smallest leg of length 3 units. The cross
sections of the solid perpendicular to that leg are semicircles. Draw a picture of the base, and a cross
section, and then find the volume
V
of the solid.
2. Let
f
(
x
) = 3
x

x
2
, and let
R
denote the region bounded by the graph of
f
and the
x
axis. Calculate
the volume
V
1
of the solid generated by revolving
R
about the
x
axis, and draw an associated figure
of the solid. Then calculate the volume
V
2
of the solid generated by revolving
R
about the
y
axis,
and draw an associated figure of the solid. From your figures, would you guess which of the volumes,
V
1
or
V
2
, should be the larger?
3. Let
f
(
x
) =
x
5
+
c/x
3
, for 1
≤
x
≤
2. We wish to find the length
L
of the graph of
f
. Determine a
positive value of
c
for which you can evaluate the integral that arises. Then find the value of
L
.
4. Suppose that when a spring is extended 6 centimeters the work
W
done is 38 ergs. How much work
W
0
is done in bringing it back to its natural length?
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 Spring '07
 Hamilton
 Right triangle, United States customary units, Let, natural length, smallest leg

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