Consider this graph of a continuous function f.

We know how to determine the area under the curve on a given interval by integrating the function on the interval.

Now, let's look at a small rectangular partition of the closed interval.

Move the slider to rotate this rectangular area about the x-axis. A cross-section of the result is shown next to the graph.

We obtain a circular disk by rotating the rectangular area about the x-axis. This is known as a solid of revolution.

We can determine the volume of this circular disk by taking the area of the circular region times the width of the rectangle.

Now, suppose we rotate the entire region on the given interval about the x-axis.

We can approximate the volume of this solid of revolution by partitioning it into n rectangles, and then summing the volume of the resulting n circular disks.

This is just a Riemann sum. Thus, the volume is given by limit of the sum as the width of the rectangle approaches zero.

This limit, in turn, is equal to the definite integral of the area function evaluated over the given interval.

Consider this function and its graph.

Consider these two functions and their graphs.

Suppose we wish to determine the volume of the figure obtained by rotating the region between the two curves on a given interval.

The volume of solid obtained by rotating the region between the curves is equal to the difference between the volumes obtained by rotating the regions under the individual curves.

Notice that a thin slice of the resulting solid is a circular ring, or "washer."

Now, consider the graph of these two functions. We can obtain a solid of revolution by rotating the given region between the curves about the x-axis.

Here we have the graphs of two curves which are both functions of y.

We can rotate the given region between the curves about the y-axis to obtain a solid of revolution. A thin slice through this solid is again a circular ring. However, this time it is centered about the y-axis.

We determine the volume of the solid of revolution in a manner similar to that used for a solid of revolution about the x-axis.

What is the figure obtained by rotating the rectangular area about the x-axis? Click the "Submit" button after selecting your answer.

How is the volume V of this solid of revolution related to those obtained by rotating the regions under each individual curve? Click the "Submit" button after selecting your answer.

What is the volume of the resulting solid of revolution? Click the "Submit" button after selecting your answer.

What is the volume of the given solid of revolution? Click the "Submit" button after selecting your answer.

What is the volume of the solid obtained by rotating the region under the curve on the given interval about the x-axis? Click the "Submit" button after selecting your answer.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education