When the region we are revolving is defined between a curve f(x) and g(x), we can find the
volume using cylindrical shells. We use the formula V = 2π x[f(x) − g(x)] dx. Here we have
a region between f(x) = x^3 and the line x = 2 that we are revolving around the line x = 2. If we
use vertical slices, then we will need to use cylindrical shells to find the volume. If we use
horizontal slices, then we will need to use washers to find the volume. Here we will use
cylindrical shells. (Try doing it yourself using washers. You should get the same answer but it
is much harder!) First, we need to find the endpoints of the region. We get the left endpoint by
setting f(x) = x^3 equal to zero and solving for x. We get x = 0. The right endpoint is simply x =
- Next, note that we are not revolving around the x-axis but around the line x = 2. Thus, the
radius of each shell is not x but rather 2 − x. The height of each shell is simply f(x) − g(x) = x^3
− 0 = x^3 . Therefore, we will find the volume by evaluating 2π [(2 − x)(x^3 )] dx = 2π (2x^3
− x^4 ) dx. We get 2π (2x^3 − x^4 ) dx = 2π = .
4.
When the region we are revolving is defined between a curve f(x) and g(x), we can find the
volume using cylindrical shells. We use the formula V = 2π x[f(x) − g(x)] dx. Here we have
f(x) = 2 (which we get by solving y^2 = 8x for y and taking the top half above the x-axis)
and g(x) = −2 (the bottom half). Thus, the height of each shell is f(x) − g(x) = 4 . We
can easily see that the left endpoint is x = 0 and that the right endpoint is x = 2. Next, note that
we are not revolving around the x-axis but around the line x = 4. Thus, the radius of each shell
is not x but rather 4 − x. Therefore, we will find the volume by evaluating: 2π
dx = 8π dx. We get 8π dx
= 8π = .
