# Find The Volume of a Frustum Using Calculus

Find the volume of a frustum using integrals and calculus.

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 = _{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 followsVolume = 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. |