Chapter 16.4, Problem 55E

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a.    An iterated integral of a function over the box D = {(x, y, z): 0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ c} can be expressed in eight different ways.

b.    One possible iterated integral of f over the prism D = {(x, y, z): 0 ≤ x ≤ 1, 0 ≤ y ≤ 3x – 3, 0 ≤ z ≤ 5} is ${\displaystyle {\int }_0^{3x-3}{\displaystyle {\int }_0^1{\displaystyle {\int }_0^{5}f\left(x,y,z\right)dzdxdy}}}$.

c.    The region $D=\left\{\left(x,y,z\right):0\le x\le 1,0\le y\le \sqrt{1-x^2},0\le z\le \sqrt{1-x^2}\right\}$ is a sphere.