Chapter 16.3, Problem 28E

To determine

Show that the integral of ${\displaystyle \underset{1}{\overset{4}{\int }}{\displaystyle \underset{0}{\overset{3}{\int }}{\displaystyle \underset{2}{\overset{6}{\int }}{x}^2{e}^{y+3z}dz}dy}dx}$ can be expressed as a product of $g(x)h(y)k(z)$ and verify:

  ${\displaystyle \underset{1}{\overset{4}{\int }}{\displaystyle \underset{0}{\overset{3}{\int }}{\displaystyle \underset{2}{\overset{6}{\int }}{x}^2{e}^{y+3z}dz}dy}dx}=\left({\displaystyle \underset{1}{\overset{4}{\int }}g\left(x\right)dx}\right)\left({\displaystyle \underset{0}{\overset{3}{\int }}h\left(y\right)dy}\right)\left({\displaystyle \underset{2}{\overset{6}{\int }}k\left(z\right)dz}\right)$