The Shell Method

This method is called the Shell Method because it uses cylindrical shells. Observe the the figure below. In this figure w is the width of the rectangle, h is the height of the rectangle, and p is the distance between the axis of revolution and the center of the rectangle. When this rectangle is revolved about its axis of revolution, it forms a cylindrical shell of thickness w. To find the volume of the shell, note the two cylinders. The radius of the larger cylinder corresponds to the outer radius of the shell, and the radius of the smaller cylinder corresponds to the inner radius of the shell. Since p is the average radius of the shell, you know the outer radius is p + (w/2) and the inner radius is p - (w/2). As a result the volume of the shell is (Volume of cylinder) - (Volume of hole).
= pi(p + (w/2))2h - pi(p - (w/2))2h
= 2piphw

Note the plane region below. You can use the above formula to find the volume of a solid of revolution. First you should assume that the plane region is revolved about a line to form the indicated solid. If you consider a horizontal rectangle of width ¤y, then as the plane region is revolved about a line parallel to the x-axis, the rectangle generates a representative shell whose volume is ¤V = 2pi[p(y)h(y)]¤y. You will then find that the limit below will give a more accurate approximation of the volume of the solid.

n
Volume of a solid = lim  2pi£  [p(yi)h(yi)]¤y
||¤||-->0      i = 1
d
= pi §  [p(y)h(y)] dy.
c

The Shell Method
To find the volume of a solid of revolution with the shell method, use on of the following.

Horizontal axis of revolution
d
Volume = V = 2pi §  p(y)h(y) dy
c

Vertical axis of revolution
b
Volume = V = 2pi §  p(x)h(x) dx
a