Find the volume of the solid generated when the region bounded by *e*^{x} and the *x*-axis on the interval [0,1] is rotated around the *y*-axis, using

(a) the washer method

(b) the shell method

Hint

Cut the solid in two at *y* = 1.

Answer

The region looks like this:

and the solid looks like this:

(a) If we try to use the washer method, we have to deal with the solid in two separate pieces. From *y* = 0 to *y* = 1, all slices perpendicular to the *y*-axis are disks of radius 1. The volume of this part of the solid is

From *y* = 1 to* y* = *e*, slices perpendicular to the *y*-axis are washers with outer radius 1 and inner radius *x* = ln *y*.

The volume of this part of the solid is

To get the volume of the entire solid, we add the volumes of its two pieces:

(b) If we use the shell method we can deal with the whole solid at once. The cylinder at position *x* has radius *x* and height y = *e*^{x}. The volume of the solid is