Volume of a solid of revolution

A slice perpendicular to the $x$-axis is a disk of radius $|f(x)|$. Integrating its area $\pi\{f(x)\}^2$ from $a$ to $b$ gives the volume.

Expand $\{f(x)\}^2$ first to get the integrand, keep $\pi$ outside, and multiply by it at the end.

Examples

1. 01Volume when rotated about the $x$-axis? (coefficient of $\pi$)$\displaystyle y=3x, \quad [0,\ 1]$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. The radius is $3x$, so a slice has area $\pi(3x)^{2}$
   2. $\displaystyle \int_{0}^{1} (3x)^{2}\,dx$$\displaystyle = 3$
   3. The question asks for the coefficient of $\pi$, so the answer is this value without the $\pi$
2. 02You have the integrand. What is this definite integral?$\displaystyle \int_{0}^{1} 4x^{2}\,dx$AnswerHide

   Answer$\displaystyle \frac{4}{3}$

   Steps

   1. $x^{2}$ integrates to $\frac{x^{3}}{3}$; put in the upper limit 1
   2. $\displaystyle \int_{0}^{1} 4x^{2}\,dx$$\displaystyle = \frac{4}{3}$
3. 03Rotated about the $x$-axis: what is the coefficient of $\pi$ in the volume?$\displaystyle y=3x, \quad [0,\ 3]$AnswerHide

   Answer$\displaystyle 81$

   Steps

   1. Add up circles of radius $3x$ from 0 to 3
   2. $\displaystyle \int_{0}^{3} (3x)^{2}\,dx$$\displaystyle = 81$
4. 04Volume when rotated about the $x$-axis? (coefficient of $\pi$)$\displaystyle y=4x, \quad [0,\ 1]$AnswerHide

   Answer$\displaystyle \frac{16}{3}$

   Steps

   1. The radius is $4x$, so a slice has area $\pi(4x)^{2}$
   2. $\displaystyle \int_{0}^{1} (4x)^{2}\,dx$$\displaystyle = \frac{16}{3}$
   3. The question asks for the coefficient of $\pi$, so the answer is this value without the $\pi$
5. 05You have the integrand. What is this definite integral?$\displaystyle \int_{0}^{1} 9x^{2}\,dx$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $x^{2}$ integrates to $\frac{x^{3}}{3}$; put in the upper limit 1
   2. $\displaystyle \int_{0}^{1} 9x^{2}\,dx$$\displaystyle = 3$
6. 06Rotated about the $x$-axis: what is the coefficient of $\pi$ in the volume?$\displaystyle y=x, \quad [0,\ 3]$AnswerHide

   Answer$\displaystyle 9$

   Steps

   1. Add up circles of radius $x$ from 0 to 3
   2. $\displaystyle \int_{0}^{3} (x)^{2}\,dx$$\displaystyle = 9$

FAQ

Q1Why is $f(x)$ squared?AnswerHide

A

Each slice is a disk of radius $|f(x)|$, and a disk has area $\pi\times(\text{radius})^2$. Squaring also makes the formula work where $f(x)$ is negative.

Q2What if the region lies between two curves?AnswerHide

A

Let $f(x)$ be the curve farther from the $x$-axis and $g(x)$ the closer one. When both are on the same side of the axis, $V=\pi\int_a^b[\{f(x)\}^2-\{g(x)\}^2]\,dx$. It is not $\pi\int_a^b\{f(x)-g(x)\}^2\,dx$.