Question

Please explain this to me!

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 $\dfrac{\pi\left(\frac d2\right)^2}2=\dfrac{\pi d^2}8$, 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

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