Question

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

the diameter of each semicircle is y. So, adding up all the discs of thickness dx, we have

v = ∫[0,1] 1/2 πr^2 dx
= π/8 ∫[0,1] e^(-2x) dx
and now it's a cinch, right?