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.

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

Volume = _{x1}^{x2}p [ f(x) ]^{ 2} dx

= _{a}^{b}p [ x^{ 2}] dy

= p [ x^{ 3} / 3] _{a}^{b}

= 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.