Finding the volume of a cube using spherical coordinates
Calculate the volume of a cube having edge length a by integrating in spherical coordinates. Suppose that the cube have all the edges on the positive semi-axis. Let us divide it by the plane passing through the points (0;0;0),(0;0;a),(a;a;0).
We get two equivalent prism; now we divide the section again by the plane passing through the points (a;0;a),(0;0;0),(a;a;a); So one may write:
So one should expect but the latter expression gives a different result. What is the correct way to calculate V and why doesn't my reasoning work?