Find the volume of the solid generated when the region bounded by ex and the x-axis on the interval [0,1] is rotated around the y-axis, using
(a) the washer method
(b) the shell method
Cut the solid in two at y = 1.
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 = ex. The volume of the solid is