Area and Volume Formulas — Question 2

Question 2 - Solution

We approximate the solid of revolution using thin disks.

Partition the interval $[a,b]$ as $$a=x_0<x_1<\cdots<x_n=b,$$ and let $\Delta x_i=x_i-x_{i-1}$. Choose a sample point $x_i^*$ in each subinterval.

Over the interval $[x_{i-1},x_i]$, the graph of $y=f(x)$ generates a disk when revolved about the $x$-axis.

The radius of this disk is $$r_i=f(x_i^*).$$

Thus, the volume of the disk is $$\Delta V_i=\pi r_i^2\,\Delta x_i =\pi [f(x_i^*)]^2\,\Delta x_i.$$

The total volume of all disks is approximated by the sum $$\sum_{i=1}^n \pi [f(x_i^*)]^2\,\Delta x_i.$$

As the partition is refined and $\|P\|\to 0$, this sum approaches the exact volume of the solid. Since $f$ is continuous, the limit exists.

Taking the limit yields $$V=\lim_{\|P\|\to 0}\sum_{i=1}^n \pi [f(x_i^*)]^2\,\Delta x_i =\pi\int_a^b [f(x)]^2\,dx.$$

$$\boxed{V=\pi\int_a^b [f(x)]^2\,dx}$$