Problem : Find the volume of a frustum with height h and radii r and R as shown below.
Solution to the problem:
A frustum may be obtained by revolving y = x between x = a and x = b around the x axis as shown below. The height h = b - a. The volume of the frustum obtained is given by
Volume = x1x2p [ f(x) ] 2 dx
= abp [ x 2] dy
= p [ x 3 / 3] ab
= p [ b 3 / 3 - a 3 / 3]
We now factor the term b 3 - a 3 and rewrite the expression for the volume as follows
Volume = p / 3 [ (b - a)(b 2 + a b + a 2 ]
We now substitute the following: h = b - a and y = x gives r = a and R = b into the expression of the volume to obtain a formula for the volume of the frustum
Volume = p / 3 [ h (R 2 + r R + r 2 ]
More references on
integrals and their applications in calculus.