All of these are examples of how the integral of x^n is 1/(n+1) x^(n+1). Combined with the fact that rescaling a n dimensional object by a factor x makes it x^n

All of these are examples of how the integral of x^n is 1/(n+1) x^(n+1). Combined with the fact that rescaling a n dimensional object by a factor x makes it x^n times bigger and that the (n+1)-volume can be expressed as an integral of n-volumes. It will generalize to any shape consisting of stacked rescaled slices of some other shape where the scaling factor increases linearly.