# What is the volume of the solid formed by the line?

## The graph of $y = 2 \sin \left(\frac{\pi x}{2}\right)$ from x = 0 to x = 2 is rotated around the y-axis to form a solid figure. Find the volume of the solid.

May 24, 2018

$\textcolor{b l u e}{V = \pi {\int}_{0}^{2} {\left(2 \sin \left(\frac{\pi x}{2}\right)\right)}^{2} \cdot \mathrm{dx} = 4 \pi}$

#### Explanation:

we will use Disk method to calculate the volume of the solid generated from rotating the curve around $\text{x-axis}$

the interval of the integral of volume $x \in \left[0 , 2\right]$

The volume of the solid between the curve and the axis is given by:

$V = \pi {\int}_{a}^{b} {y}^{2} \cdot \mathrm{dx}$

$V = \pi {\int}_{0}^{2} {\left(2 \sin \left(\frac{\pi x}{2}\right)\right)}^{2} \cdot \mathrm{dx}$

$V = \pi {\int}_{0}^{2} \left(4 {\sin}^{2} \left(\frac{\pi x}{2}\right)\right) \cdot \mathrm{dx}$

$V = 4 \pi {\int}_{0}^{2} \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\pi x\right)\right) \cdot \mathrm{dx}$

$V = 2 \pi {\int}_{0}^{2} \left(1 - \cos \left(\pi x\right)\right) \cdot \mathrm{dx}$

$= 2 \pi {\left[x - \frac{1}{\pi} \sin \left(\pi x\right)\right]}_{0}^{2} = \left[\left(2 - \frac{1}{\pi} \cdot \sin 2 \pi\right) - \left(- \frac{1}{\pi} \sin 0\right)\right]$

$2 - 0 + 0 = 4 \pi$