346
6.2. USING DEFINITE INTEGRALS TO FIND VOLUME
since the volume of a cylinder of radius r and height h is V = πr 2 h.
Using a definite integral to sum the volumes of the representative slices, it follows that
V =
2
−2
π(4 − x
2 )
2 dx.
It is straightforward to evaluate the integral and find that the volume is V =
512
15 π.
For a solid such as the one in Example 6.1, where each cross-section is a cylindrical
disk, we first find the volume of a typical cross-section (noting particularly how this volume
depends on x), and then we integrate over the range of x-values through which we slice
the solid in order to find the exact total volume. Often, we will be content with simply
finding the integral that represents the sought volume; if we desire a numeric value for the
integral, we typically use a calculator or computer algebra system to find that value.
The general principle we are using to find the volume of a solid of revolution generated
by a single curve is often called the disk method.
If y = r(x) is a nonnegative continuous function on [a, b], then the volume of the
solid of revolution generated by revolving the curve about the x-axis over this interval
is given by
V =
b
a
πr(x)
2 dx.
A different type of solid can emerge when two curves are involved, as we see in the
following example.
Example 6.2. Find the volume of the solid of revolution generated when the finite region
R that lies between y = 4 − x 2 and y = x + 2 is revolved about the x-axis.
Solution.
First, we must determine where the curves y = 4 − x 2 and y = x + 2 intersect.
Substituting the expression for y from the second equation into the first equation, we find
that x + 2 = 4 − x 2 . Rearranging, it follows that
x
2 + x − 2 = 0,
and the solutions to this equation are x = −2 and x = 1. The curves therefore cross at
(−2, 0) and (1, 1).
6.2. USING DEFINITE INTEGRALS TO FIND VOLUME
since the volume of a cylinder of radius r and height h is V = πr 2 h.
Using a definite integral to sum the volumes of the representative slices, it follows that
V =
2
−2
π(4 − x
2 )
2 dx.
It is straightforward to evaluate the integral and find that the volume is V =
512
15 π.
For a solid such as the one in Example 6.1, where each cross-section is a cylindrical
disk, we first find the volume of a typical cross-section (noting particularly how this volume
depends on x), and then we integrate over the range of x-values through which we slice
the solid in order to find the exact total volume. Often, we will be content with simply
finding the integral that represents the sought volume; if we desire a numeric value for the
integral, we typically use a calculator or computer algebra system to find that value.
The general principle we are using to find the volume of a solid of revolution generated
by a single curve is often called the disk method.
If y = r(x) is a nonnegative continuous function on [a, b], then the volume of the
solid of revolution generated by revolving the curve about the x-axis over this interval
is given by
V =
b
a
πr(x)
2 dx.
A different type of solid can emerge when two curves are involved, as we see in the
following example.
Example 6.2. Find the volume of the solid of revolution generated when the finite region
R that lies between y = 4 − x 2 and y = x + 2 is revolved about the x-axis.
Solution.
First, we must determine where the curves y = 4 − x 2 and y = x + 2 intersect.
Substituting the expression for y from the second equation into the first equation, we find
that x + 2 = 4 − x 2 . Rearranging, it follows that
x
2 + x − 2 = 0,
and the solutions to this equation are x = −2 and x = 1. The curves therefore cross at
(−2, 0) and (1, 1).