Here you will need strong visualization skills. The inequality describes asolidsphere with radius centered on the origin. If , then we are only talking about the top half of the figure, the part above the xy-plane, a "semi-sphere". This shape is thedomainfor thefunction\to\mathbb{R}" alt="r\to\mathbb{R}" />, not defined in your problem. So we are actually finding the hypervolume of a 4-dimensional shape.

The bounds on go from to , as given in the problem, so that will be the innermost integral. So we are taking horizontal cross-sections of the semi-sphere, giving us circular disks of radius . So must range from to . The intersection of these two planes give a line of length . Therefore the outermost integral will be ranging from to . Hence,

(I suppose it would be inappropriate in this particular problem to convert to polar or cylindrical coordinates, which would incredibly simplify the problem.)