Note that it is enough to consider the case of the unit sphere, since to get an ellipsoid from the unit sphere you just have to apply a diagonal matrix whose entries are the semi axis lengths, which multiplies the volume by the determinant, I.e., the product of the semi axis lengths. Also,it is easy to get the volume of the n-sphere inductively, with just one easy integration by slicing it by planes parallel to an axis and noting that the slices are spheres of one lower dimension and easily computed radii.