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Question

# Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y=cos 2x, y=0, x=0, x=π/4

Solution

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To find the $\textbf{Volume}$ of the solid given we will apply $\textbf{The Disk Method}$ ...

$\boxed{\color{#c34632}{V=\pi\int_{a}^{b}{R(x)^2dx}}}$

From the exercise given we can see that we need to find

$$$V=\pi\int_{0}^{\pi/4}{(\cos{(2x)})^2dx}$$$

$\textbf{This leads}$ to:

\begin{align*} V&=\pi\int_{0}^{\pi/4}{\frac{\cos{(4x)}-1}{2}dx}\\ &=\frac{\pi}{2}\int_{0}^{\pi/4}{(\cos{(4x)}-1)dx}\\ &=\frac{\pi}{2}\left[\frac{\sin{(4x)}}{4}+x\right]_{0}^{\pi/4}\\ &=\frac{\pi^2}{8}\\ \end{align*}

So, our $\textbf{final answer}$ is:

$\boxed{\color{#c34632}{\frac{\pi^2}{8}}}$

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