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.


\( \begin{align} \displaystyle
x^2 &= \sqrt{x} \\
x^4 &= x \\
x^4 – x &= 0 \\
x(x^3-1) &= 0 \\
x &= 0 \text{ and } x=1 \\
V &= \pi \int_{0}^{1}{\big[\sqrt{x}^2 – (x^2)^2\big]}dx \\
&= \pi \int_{0}^{1}{(x – x^4)}dx \\
&= \pi \bigg[\dfrac{x^2}{2} – \dfrac{x^5}{5}\bigg]_{0}^{1} \\
&= \pi \bigg[\dfrac{1^2}{2} – \dfrac{1^5}{5}\bigg] – \pi \bigg[\dfrac{0^2}{2} – \dfrac{0^5}{5}\bigg] \\
&= \dfrac{3 \pi}{10} \text{ units}^3
\end{align} \)


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