# Archimedes Revenge Solution

^{2}+1) (π/2-2 arctan(z)) + z

^{2}-1 ]/2 from 0 to 1. (The third picture shows one of those regions). To solve the integral, note that the anti-derivative of arctan(z) is x arctan(x) - log(1+x^2)/2 (integration by parts). The answer of 1/2 + 3 I, which is log(2). Since there are 8 pieces, the volume of our beautiful solid is 8 log(2) = log(256). Update June 27, 2021: a much better, clearer and more detailed solution was written down by Mark Villarino Bertram: