MATH 152 - Recitation Quiz 3 - Spring 2015
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1. Find the volume of the solid generated by rotating the region bounded by the given curves about the
y -axis.
√
y = sin x2 , y = 0, x = 0, x =
π
Since we are rotating about the y -axis, we will use the Shell Method to nd the volume of the solid. To do
this we draw a vertical rectangle in our region (in other words, one that is parallel to the y -axis). Remember
that the area of our region is then 2πrh. The radius of our shell is the distance between the rectangle and
the axis of revolution. Therefore,
r (x) = x. The height of our shell is the height of the rectangle. Therefore,
the height is h (x) = sin x2 . Therefore, the following gives the volume of the solid:
√
ˆπ
2πx sin x2 dx.
V =
0
We will evaluate the integral using substitution. Let u = x2 . Then du = 2xdx. Therefore,
ˆπ
(sin u) du = −π [cos u]0 = −π [cos π − cos 0] = −π (−1 − 1) = 2π units3 .
π
V =π
0
2. A spring has a natural length of 20 cm. If a 25-N force is required to keep it stretched to a length of
30 cm, write (but do not evaluate) an integral expression which yields the work required to stretch the
spring from 20 cm long to 25 cm long.
Hook's Law tells us that the force used to stretch the spring x meters from its natural length is given by
f (x) = kx, where k is some constant. A force of 25 N is applied in order to stretch the spring to 30 cm (which
is 0.1 m longer than its natural length). Therefore, 25 = k · 0.1 =⇒ k = 250. Therefore, f (x) = 250x. In
order to nd the total work involved in stretching the spring from 20 cm to 25 cm (a total of 0.05 m passed
its natural length), we would need to evaluate the following integral:
0.05
ˆ
W =
250xdx.
0
3. Write (but do not evaluate) an integral expression which yields the average value of the given function
on the given interval:
2
f (x) = ex , [1, 4] .
In order to nd the average value of the function on the interval, we have to integrate the function over the
interval and then divide by the length of the interval. Therefore,
fave
1
=
4−1
ˆ4
1
1
1
e dx =
3
x2
ˆ4
2
ex dx.
1