Let be a function continuous on an interval . The volume of the solid of revolution obtained by rotating about the -axis on the interval is given by
for every . Find a formula for the function .
Using the formula for the volume of the solid of revolution generated by a function on an interval we know
Now we differentiate both sides of this equation using the fundamental theorem of calculus on the right-hand side,
Sketch the graph of region between these functions and compute the volume of the solid of revolution generated by revolving this region about the -axis.
The sketch of the region between and is as follows:
We then compute the volume of the solid of revolution as follows.