# Find The Volume of a Frustum Using Calculus

Use the method of the disk around the x axis to find the volume of a frustum.

## Volume of Frustum Formula

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 $$y = m x$$ between $$x = a$$ and $$x = b$$ around the x axis as shown below. The height $$h = b - a$$.

Rotating a disk (red) of radius $$y$$ hence of area $$\pi y^2$$ and thikness $$\Delta x$$, the volume $$V$$ of the frustum may be written as
$V = \int_a^b \pi y^2 dx \quad (I)$ The slope $$m$$ is given by $m = \dfrac{R - r}{h}$ where $$h$$ is the height of the frustum given by $h = b - a$ Substitute $$y$$ by $$mx$$ in (I) and write $V = \displaystyle m^2 \pi \int_a^b x^2 dx$ Evaluate the integral $V = m^2 \pi \left[\dfrac{1}{3} x^3 \right]_a^b$ $\qquad = \dfrac{1}{3} m^2 \pi (b^3 - a^3) \quad (II)$ Note that $r = m \; a$ and $R = m \; b$ Hence $a = \dfrac{r}{m}$ and $b = \dfrac{R}{m}$ Substitute in (II) $\qquad V = \dfrac{1}{3} m^2 \pi \left(\left(\dfrac{R}{m}\right)^3 - \left(\dfrac{r}{m}\right)^3\right)$ Simplify $V = \dfrac{1}{3 \; m} \pi \left(R^3 - r^3\right)$ Substitute $$m$$ by $$\dfrac{R - r}{h}$$ in the above and rewrite as $V = \dfrac{ \pi h}{3} \dfrac{ \left(R^3 - r^3\right)}{R-r} \quad (III)$ Note that using division of polynomials in two variables, $$\dfrac{\left(R^3 - r^3\right)}{R-r}$$ may be simplified as $\dfrac{\left(R^3 - r^3\right)}{R-r} = R^2 + r R + r^2$ We now substitute the above in (III) to obtain the final formula for the volume of the frustum $\boxed {V = \dfrac{\pi h}{3} \left( R^2 + r R + r^2 \right) }$