Chapter 15.6, Problem 28E

Midpoint Rule for Triple Integrals In the Midpoint Rule for triple integrals we use a triple Riemann sum to approximate a triple integral over a box $B$ , where $f(x,y,z)$ is evaluated at the center $\left({\bar{x}}_i,{\bar{y}}_j,{\bar{z}}_k\right)$ of the box $B_{ijk}$ . Use the Midpoint Rule to estimate the value of the integral. Divide $B$ into eight sub-boxes of equal size.

28. ${\displaystyle {\iiint }_B\sqrt{x^2+y^2+z^2}}dV$ , where

$B=\{(x,y,z)\mid 0⩽x⩽4,0⩽y⩽4,0⩽z⩽4\}$