# How do you find the volume of the solid generated by revolving the region bounded by the graphs y=e^(x/2), y=0, x=0, x=4, about the x axis?

Dec 15, 2016

#### Explanation:

Here is a graph of the region in blue. A slice has been taken perpendicular to the axis of rotation. The rotation as shown by the arrow/arc.

The representative slice is a disc of

thickness dx

$r = {y}_{\text{greater" - y_"lesser}} = {e}^{\frac{x}{2}} - 0 = {e}^{\frac{x}{2}}$ .

The volume of the representative slice (disc) is

$\pi {r}^{2} \text{thickness} = \pi {\left({e}^{\frac{x}{2}}\right)}^{2} \mathrm{dx} = \pi {e}^{x} \mathrm{dx}$ .

The values of $x$ vary from $0$ to $4$, so the resulting solid has volume

$V = {\int}_{0}^{4} \pi {e}^{x} \mathrm{dx}$.

Evaluate the integral to get

$V = \pi \left({e}^{4} - 1\right) \approx 53.6$ (cubic units).