Micaela Simon

2022-06-30

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

Belen Bentley

Beginner2022-07-01Added 28 answers

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 $\pi {r}^{2}$, 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 $={\int}_{a}^{b}\pi f(x{)}^{2}dx=\pi {\int}_{a}^{b}f(x{)}^{2}dx$

The area of a circle is $\pi {r}^{2}$, 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 $={\int}_{a}^{b}\pi f(x{)}^{2}dx=\pi {\int}_{a}^{b}f(x{)}^{2}dx$

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