Let $y=f(x)$ and $y=g(x)$ be two curves in the positive $xy$-coordinate with intersection points at $x=1$ and $x=4$. Let $R$ be a region in the positive $xy$-coordinate plane bounded by the curves $y=f(x)$ and $y=g(x)$ (where $f(x)\ge g(x)$ everywhere in the region) . Let $V$ be the volume of solid whose base is $R$ and cross-sections perpendicular to $x$-axis are semi-circles. How to set up $V$ as an integral?
My try: Thinking of splitting the volume into semi-cylindrical slabs of length $\Delta x$, the volume of each semi-cylindrical slab should be $\dfrac 12 \pi (f(x)-g(x))^2\Delta x$. So the volume $V$ should be $\int_{1}^4 \dfrac 12 \pi (f(x)-g(x))^2 dx $. Is my answer right? Especially, are my bounds of integration correct?