What you need to do here is to slice your shape into discs with lots of planes perpendicular to the $x$-axis, so that each disc has width $\delta x$. The volume of a disc with left-hand end-point $x$, and right-hand end-point $x+\delta x$ is approximately $(\delta x)\pi e^{4x}$, since the volume of a cylinder is $\pi r^{2}h$.

Now, in order to calculate the total volume, we need to add up the volumes of these discs; we turn this finite sum into an integral by letting $\delta x$ tend to zero. If this doesn't make much sense, think of approximating the area under a curve by rectangles of width $\delta x$ and letting $\delta x$ tend to zero: we end up with an integral.

Thus the total volume is

${\displaystyle {\int }_0^{1/2}\pi e^{4x}\mathrm{dx}}$

which I will leave you to calculate.

Hope this helps.