Volumes for Two Functions

If the region bounded by the upper function $y_{upper}=f(x)$ and the lower funciton $y_{lower}=g(x)$, and the lines $x=a$ and $x=b$ is revolved about the $x$-axis, then its volume of revolution is given by:
$$ \begin{align} \displaystyle
V &= \int_{a}^{b}{\Big([f(x)]^2 – [g(x)]^2\Big)}dx \\
&= \int_{a}^{b}{\Big(y_{upper}^2 – y_{lower}^2\Big)}dx
\end{align} $$

Example 1

Find the volume of revolution generated by revolving the region between $y=x^2$ and $y=\sqrt{x}$ about the $x$-axis.

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