Regions of the n-sphere and related integrals

In this note the volumes of certain regions in the n-sphere will be found in two ways: (a) by using a symmetry argument, (b) by expressing the volumes as repeated integrals over the (n-l)-cube. By considering the 4 and 5 spheres and equating the integrals obtained by method (b) to the solution obtained by method (a) we evaluate integrals of the form for certain values of a, b and c; it does not appear easy (if indeed it is possible) to evaluate these integrals by direct methods.