Chapter 15.6, Problem 24E

(a) In the Midpoint Rule for triple integrals we use a triple Riemann sum to approximate a triple integral over a box B, where $f\left(x,y,z\right)$  is evaluated at the center $\left(\overline{x_i},\overline{y_j},\overline{z_k}\right)$ of the box $B_{ijk}$. Use the Midpoint Rule to estimate ${\displaystyle {\iiint }_B\sqrt{x^2+y^2+z^2}}dV$, where B is the cube defined by $0\le x\le 4,0\le y\le 4,0\le z\le 4$. Divide B into eight cubes of equal size.

(b) Use a computer algebra system to approximate the integral in part (a) correct to the nearest integer. Compare with the answer to part (a).