# Math 12 -- Calculus II

**Santa Clara University**

**Department of Mathematics and
Computer Science**

**Dennis C. Smolarski,
S.J.**

Math 12 Homepage (Smolarski)
## Volumes and Surfaces of Rotation

Calculus can be used to find volumes of solids that have been created
by rotating a two-dimensional curve around either axis or any
arbitrary line parallel to either axis. It is
easy to acknowledge that the cross section of the
resulting solid is circular, but what is often more difficult
to determine is the exact shape of the solid.
These graphs and animations
are meant to help a person visual the resulting solid.

### Parabola Example

The following is a graph of the parabola, *y = x*^{2} + 1.

### Rotation Around the *x*-Axis

If we rotate this figure around the *x-*axis, we see the
three-dimensional surface that it creates.

### Rotation Around the *y*-Axis

If we rotate this figure (or the positive side of the
figure, i.e., when *x* is greater than zero),
around the *y-*axis, we get a
figure usually called a paraboloid, which looks like the nose-cone
of a rocket.
We see this figure being drawn out in the following animation.

The complete figure is perhaps better visualized in this
animation.

**NOTE:** The animation can be stopped at any
time by pressing the **escape** key, and restarted by clicking
on the **Reload** button of the browser.

Last updated: 13 January 2001.
Maintained by Dennis C.
Smolarski, SJ.