5128_Ch07_pp378-433

Section 7.3 Volumes 403
2. Identify the limits of integration: y runs from 0 to 2.
3. Integrate to find the volume.
V 2
0
2
0
2p( radius)( shell
height) shell dy
2py4 y 2 dy 8p
Now try Exercise 33(a).
4
2
y
0 2
Figure 7.28 The region and the height
of a typical shell in Example 5.
o
x k
[–1, 3.5] by [–0.8, 2.2]
Figure 7.29 The base of the paperweight
in Example 6. The segment perpendicular
to the x-axis at x k is the diameter of
a semicircle that is perpendicular to the
base.
0
2
y
x
x
π
y = 2 sin x
Figure 7.30 Cross sections perpendicular
to the region in Figure 7.29 are semicircular.
(Example 6)
EXAMPLE 5
Finding Volumes Using Cylindrical Shells
The region bounded by the curves y 4 x 2 , y x, and x 0 is revolved about the
y-axis to form a solid. Use cylindrical shells to find the volume of the solid.
SOLUTION
1. Sketch the region and draw a line segment across it parallel to the y-axis
(Figure 7.28). The segment’s length (shell height) is 4 x 2 x. The distance of the
segment from the axis of revolution (shell radius) is x.
2. Identify the limits of integration: The x-coordinate of the point of intersection of the
curves y 4 x 2 and y x in the first quadrant is about 1.562. So x runs from 0 to
1.562.
3. Integrate to find the volume.
Other Cross Sections
radius)( height) shell dx
1.562
2p(x)(4 x 2 x) dx
0
13.327
1.562
V 2p ( shell
0
Now try Exercise 35.
The method of cross-section slicing can be used to find volumes of a wide variety of unusually
shaped solids, so long as the cross sections have areas that we can describe with
some formula. Admittedly, it does take a special artistic talent to draw some of these
solids, but a crude picture is usually enough to suggest how to set up the integral.
EXAMPLE 6
A Mathematician’s Paperweight
A mathematician has a paperweight made so that its base is the shape of the region between
the x-axis and one arch of the curve y 2 sin x (linear units in inches). Each cross
section cut perpendicular to the x-axis (and hence to the xy-plane) is a semicircle whose
diameter runs from the x-axis to the curve. (Think of the cross section as a semicircular
fin sticking up out of the plane.) Find the volume of the paperweight.
SOLUTION
The paperweight is not easily drawn, but we know what it looks like. Its base is the region
in Figure 7.29, and the cross sections perpendicular to the base are semicircular fins like
those in Figure 7.30.
The semicircle at each point x has
radius 2s in x
sin x and area Ax 1 2
2 psin x2 .
continued