However, there is a class of solids for

which our single variable calculus will work.

These are the volumes of revolution,

that is solids obtained by rotating some two dimensional shape

about an axis, sweeping out a three dimensional volume.

There are two principle ways to decompose such an object

into simple volume elements.

The first is by taking slices orthogonal to

the axis of revolution and progressing along the axis.

This tends to lead to disc-like volume elements.

Of course, the other way is parallel to the axis of revolution.

Slicing about a region that is parallel

leads to a cylindrical volume element in general.

Let's do some examples to see these two approaches..

Let us consider this solid formed by rotating a disk about an axis.

One might describe this object as doughnut like.

What's the volume of such an object?

Let's set up coordinates so that the Y axis is the axis of revolution.

And that the disc that is rotated there about is of radius a.

And the center of the disc is located

a distance capital R away from the axis of rotation.

Then, decomposing this volume by slices that are parallel

to the axis of revolution, gives us a cylindrical volume element.

Let's compute that volume element and then integrate.

The volume element intersects this disk along a vertical strip.

What is the height of that vertical strip if we use

x to denote our distance from the axis of rotation?

Then by building the appropriate right triangle and

recalling that this disc has radius a, we see that the height

of this cylinder is twice the square root of a squared- quantity

(x-R) quantity squared.

Thus the volume element, being cylindrical, is what?

We have to take the circumference, that is 2 pi x, and

multiply it by the height, twice root a squared- (x-R) squared,

and then multiply that by the thickness, dx.