Volumes using Integration

Volume of Revolution

We can use integration to find volumes of revolution between $x=a$ and $x=b$.
When the region enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$ is revolved through $2 \pi$ or $360^{\circ}$about the $x$-axis to generate a solid, the volume of the solid is given by: $$ \begin{align} \displaystyle V &= \lim_{h \rightarrow 0} \sum_{x=a}^{x=b}{\pi \big[f(x)\big]^2 h} \\ &= \int_{a}^{b}{\pi \big[f(x)\big]^2}dx \\ &= \pi \int_{a}^{b}{y^2}dx \end{align} $$

Example 1

Find the volume of the solid generated when the line $y=x$ for $1 \le x \le 3$ is rovolved through $2 \pi$ or $360^{\circ}$ around the $x$-axis.

Example 2

Find the volume of the solid generated when the line $y=\sqrt{x}$ for $0 \le x \le 2$ is rovolved through $2 \pi$ or $360^{\circ}$ around the $x$-axis.