Page 82 - Calc for the Clueless

Note
If we noticed that the two regious have the same area, we could have found the area of either one
and doubled its value.

Volumes of Rotations

The next topic is finding volumes of rotations. This is very visual. If you see the picture, the volume is easy. If
not, this topic is very hard.

Imagine a perfectly formed apple with a line through the middle from top to bottom. We can find the volume
two different ways. One way is by making slices perpendicular to the line (axis). (We will do the other way
later with an onion.) Each slice is a disc, a thin cylinder. Its volume is πr h, where h is very small. If we add up
2
all the discs, taking the limits properly, we get the volume.

We will take the same region in six different problems, rotating this region differently six times and getting six
different volumes.

Example 5—

Find the volume if the region R is rotated about the x axis.

2
2
2
The volume of each disc is πr h. h = ∆x. r = y. So r = y = x; x goes from 0 to 9.

The integrals are almost always easy. Once you understand the picture, all will be easy. But it takes most people
time to study the pictures.

Let's get back to our apple. Suppose we core our apple. When we take slices perpendicular to the axis, we get
rings. The area of a ring is the area of the outside minus the area of the inside. The volume of each disc is