Find The Volume of a Frustum Using Calculus

Find the volume of a frustum using integrals and calculus.

Problem : Find the volume of a frustum with height h and radii r and R as shown below.

frustum used in problem


Solution to the problem:

A frustrum 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

frustum of probelm above in system of axes


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.