Question

Please explain this to me!

The base of a solid in the region bounded by the graphs of y = e-x y = 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

I've attached a plot of one such cross section (grayish orange) with its diameter lying in the region (blue) bounded above by the curve
`y=e^(-x)` (red). (This cross section is taken at
`x=0.25`.)

The area of each cross section in terms of its diameter
`d` is
`\frac{\pi\left(\frac d2\right)^2}2=\frac{\pi d^2}8`, where the diameter is determined by the vertical distance (in the x-y plane) between the curve
`y=e^(-x)` and the x-axis
`y=0`. So,
`d=e^(-x)`.

The volume would then be


`\displaystyle\int_0^1\frac{\pi(e^(-x))^2}8\,\mathrm dx=\frac\pi8\int_0^1e^(-2x)\,\mathrm dx=(\pi(e^2-1))/(16e^2)`