f(x) = and the x-axis. First, we need to find the endpoints of the region. We do this by
setting f(x) = equal to zero and solving for x. We get x = −3 and x = 3. Thus, we will
find the volume by evaluating dx = π (9 − x^2 ) dx. We get π (9 − x^2 ) dx
= π = 36π.
- 2 π
When the region we are revolving is defined between a curve f(x) and the x-axis, we can find
the volume using disks. We use the formula V = π [f(x)]^2 dx. Here we have a region between
f(x) = sec x and the x-axis. We are given the endpoints of the region: x = − and x = . Thus,
we will find the volume by evaluating π sec^2 x dx. We get π sec^2 x dx = π(tan x) =
2 π.
3.
When the region we are revolving is defined between a curve f(y) and the y-axis, we can find
the volume using disks. We use the formula V = π [f(y)]^2 dy (see this page and note that when
g(y) = 0 we get disks instead of washers). Here we have a region between f(y) = 1 − y^2 and the
y-axis. First, we need to find the endpoints of the region. We do this by setting f(y) = 1 − y^2
equal to zero and solving for y. We get y = −1 and y = 1.
Thus, we will find the volume by evaluating π (1− y^2 )^2 dy = π (1 − 2y^2 + y^4 ) dy. We get π
(1 − 2y^2 + y^4 ) dy = π.
- 2 π
When the region we are revolving is defined between a curve f(y) and the y-axis, we can find
the volume using disks. We use the formula V = π [f(y)]^2 dy (see this page and note that when
g(y) = 0 we get disks instead of washers). Here we have a region between f(y) = y^2 and the
y-axis. We are given the endpoints of the region: y = −1 and y = 1. Thus, we will find the
