Chapter 16.3, Problem 27E

To determine

Show that the integral of f over box $B=\left[a,b\right]\times \left[c,d\right]\times \left[p,q\right]$ , where $f(x,y,z)=g(x)h(y)k(z)$ can be expressed as a product of integral as follow:

  ${\displaystyle \underset{B}{\iiint }f\left(x,y,z\right)dV}=\left({\displaystyle \underset{a}{\overset{b}{\int }}g\left(x\right)dx}\right)\left({\displaystyle \underset{c}{\overset{d}{\int }}g\left(y\right)dy}\right)\left({\displaystyle \underset{p}{\overset{q}{\int }}g\left(z\right)dz}\right)$