Find the largest cylinder (in terms of volume) we can obtain by revolving a rectangle lying on the -axis and contained entirely in the region of the plane bounded by the axes and the curve

First, we know that the two upper corners must have the same distance from the -axis (since this is a rectangle the line joining the upper corners must be horizontal). Both upper corners will lie on the curve since if either corner were not on the curve we could obtain a larger rectangle (hence a larger cylinder when we rotate) by extending the rectangle so the corner is on the curve. Let be the the coordinate of the upper left corner, and be the coordinate of the upper right corner. Since these both lie on the curve we have,

However, one of these solutions, is the trivial case that , so the cylinder in this case has volume 0. The other solution then is

So, then we compute the volume of the cylinder,

Taking the derivative,

Setting this derivative equal to 0, we then have

These two values of then correspond to the values for and , so we have

(Where we can eliminate the negative value of since the problem requires the value to be positive.) Therefore, the maximum volume is given by