Question

The base of a solid in the region bounded by the graphs of y=e^-xy=0, and x=0, and x=1. cross sections of the solid perpendicular to the x-axis are semicircles. what is the volume, in cubic units, of the solid?

Answer 1

The diameter of each semicircular cross-section is given by the vertical distance between the curves $y=e^{-x}$ and $y=0$, which is just $e^{-x}$.

Given a semicircle with diameter $d$, its area is $\dfrac{\pi d^2}8$, so the area of each semicircular section is $\dfrac{\pi e^{-2x}}8$.

The volume of the solid is obtained by integrating from 0 to 1:

$\displaystyle\frac\pi8\int_{x=0}^{x=1}e^{-2x}\,\mathrm dx=\frac{(e^2-1)\pi}{16e^2}$