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 = x2 + 1.

[Maple Plot]

Rotation Around the x-Axis

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

[Maple Plot]

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.

[Maple Plot]

The complete figure is perhaps better visualized in this animation.

[Maple Plot]

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.