Quick Circle Cross Sections Question So I was given the following prompt:

"Let $R$ be the region in the first quadrant enclosed by the graph of $f(x)=\sqrt{\cos(x)}$, the graph of $g(x)=e^x$, and the vertical line $x=\frac {\pi}{2}$, as shown in the figure below. Region $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are semicircles with diameters on the $xy$-plane. Write, but do not evaluate, an integral expression that gives the volume of this solid."


I'm a bit confused about how I'd go about even starting a problem like this. I understand that in order to find the volume of a solid like this that I'd have to integrate the area formula with some bounds, but I'm a bit lost about what this equation would look like here. Any help would be appreciated!
 A: The diameter is between the two given curves so the radius of the semicircle is
$r = \displaystyle \frac{e^x - \sqrt{\cos x}}{2} $
Bound of $x$ is $0 \leq x \leq \frac{\pi}{2}$. Finally the range of $\theta$ is $\pi$ as cross-sections are semicircles.
So the integral is $\displaystyle \int_0^\pi \int_0^{\pi/2} \int_0^{\frac{e^x - \sqrt{\cos x}}{2}} r \ dr \ dx \ d\theta$
A: Since you know that the cross sections are semi-circles, you can think of splitting the volume into semi-cylindrical slabs of length $\Delta x$.  The radius of each semi-cylindrical slab is $$r = \frac{e^{x} - \sqrt{\cos(x)}}{2},$$ and so the volume is $$\frac{1}{2}\pi r^2 \Delta x = \frac{1}{2}\pi \left(\frac{e^{x} - \sqrt{\cos(x)}}{2}\right)^{2}\Delta x = \frac{1}{8}\pi \left(e^{x} - \sqrt{\cos(x)}\right)^{2}\Delta x.$$  Writing the volume as a Riemann sum:
$$V \approx \sum_{i=1}^{n} \frac{1}{8}\pi \left(e^{x_{i}} - \sqrt{\cos(x_{i})}\right)^{2}\Delta x_{i},$$ and upon taking the limit as $n\to \infty$, we get the integral $$V = \frac{\pi}{8}\int_{0}^{\frac{\pi}{2}}\left(e^{x} - \sqrt{\cos(x)}\right)^{2}\,dx.$$
