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

Question

in cubic units of the solid

Answer 1

Calculate the radius as a function of x: r = 1/2e^-x

The area of each cross section is 1/2πr²

So the volume = ₀∫¹(1/2πr²) dx = ₀∫¹(1/2π(1/2e^-x)²) dx = π/8•₀∫¹ e^-2x dx

You can use a calculator if you want to calculate the integral but it's fairly easy to do manually.