I need some help with the working and solution to this volume and integration problem. I don't really have a starting point.
The region bounded by $y=-1$, $y=e^{2x}$, $x=0$ and $x=2$ is revolved about the line $y=-1$. Find the volume of the resulting solid.
On
Use the disk method to integrate over $x$. For any $x$ within $[0,2]$, the disk radius is $e^{2x}+1$.
Thus, the volume integral is,
$$V=\pi \int_0^2 (e^{2x}+1)^2dx$$ $$=\pi \int_0^2 (e^{4x}+2e^{2x}+1)dx$$ $$=\pi \left(\frac14( e^{8}-1)+(e^{4}-1)+2\right)$$ $$=\frac\pi4 \left( e^{8}+4e^{4}+3\right)$$
Hint: Do a change of variables so that you are revolving the solid about the line $y=0$, IE the x-axis, instead.