Question

How do you determine the volume of a solid created by revolving a function around an axis?

Answer 1

`"Volume" = pi int_a^b f(x)^2 dx`

Explanation:

Given a function `f(x)` and an interval `[a, b]` we can think of the solid formed by revolving the graph of `f(x)` around the `x` axis as a horizontal stack of an infinite number of infinitesimally thin disks, each of radius `f(x)`.

The area of a circle is `pir^2`, so the area of the circle at a point `x` will be `pi f(x)^2`.

The volume of the solid is then the infinite sum of the infinitesimally thin disks over the interval `[a, b]`

So:

`"Volume" = int_a^b pif(x)^2 dx = pi int_a^b f(x)^2 dx`