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Section 15.6: Triple Integration

Exercise 24

  1. 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 \((\bar{x}_i, \bar{y}_j, \bar{z}_k)\) of the box \(B_{ijk}\). Use the Midpoint Rule to estimate \(\iiint_B \sqrt{x^2 + y^2 + z^2} dV\), 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.
  2. Use a computer algebra system to approximate the integral in part (a) correct to the nearest integer. Compare with the answer to part (a).