5128_Ch07_pp378-433

Section 7.3 Volumes 399
7.3
Volumes
What you’ll learn about
• Volume As an Integral
• Square Cross Sections
• Circular Cross Sections
• Cylindrical Shells
• Other Cross Sections
. . . and why
The techniques of this section
allow us to compute volumes of
certain solids in three dimensions.
y
S
P x
Cross-section R(x)
with area A(x)
Volume As an Integral
In Section 5.1, Example 3, we estimated the volume of a sphere by partitioning it into thin
slices that were nearly cylindrical and summing the cylinders’ volumes using MRAM.
MRAM sums are Riemann sums, and had we known how at the time, we could have continued
on to express the volume of the sphere as a definite integral.
Starting the same way, we can now find the volumes of a great many solids by integration.
Suppose we want to find the volume of a solid like the one in Figure 7.15. The cross
section of the solid at each point x in the interval a, b is a region Rx of area Ax. If A is a
continuous function of x, we can use it to define and calculate the volume of the solid as an
integral in the following way.
We partition a, b into subintervals of length Δx and slice the solid, as we would a loaf
of bread, by planes perpendicular to the x-axis at the partition points. The kth slice, the
one between the planes at x k1 and x k , has approximately the same volume as the cylinder
between the two planes based on the region Rx k (Figure 7.16).
y
Plane at x k1
Approximating
cylinder based
on R(x k ) has height
Δx k x k x k1
0
a
x
0
x k1
Plane at x k
Figure 7.15 The cross section of an
arbitrary solid at point x.
b
x
The cylinder’s base
is the region R(x k )
with area A(x k )
NOT TO SCALE
x k
x
Figure 7.16 Enlarged view of the slice of the solid between the planes at x k1 and x k .
The volume of the cylinder is
V k base area height Ax k Δx.
The sum
V k Ax k Δx
approximates the volume of the solid.
This is a Riemann sum for Ax on a, b. We expect the approximations to improve as
the norms of the partitions go to zero, so we define their limiting integral to be the volume
of the solid.
DEFINITION
Volume of a Solid
The volume of a solid of known integrable cross section area Ax from x a to
x b is the integral of A from a to b,
V b
a
Ax dx.