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
`\frac{\pi d^2}8`, so the area of each semicircular section is
`\frac{\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=((e^2-1)\pi)/(16e^2)`