Appendix: Integrations Over the 2-Spheres

This is done by using Simpson's rule twice. Once in each coordinate direction. Simpson's rule is

$$\int^{x_2}_{x_1} f(x) dx = \frac{h}{3} [f_1+4f_2+2f_3+4f_4+\ldots+2f_{N-2}+4 f_{N-1}+f_N] +O(1/N^4)$$ (part3132)

$N$ must be an odd number.