Example: Compute the integral
$\iint_R \frac{4}{3}(5x) dA$, where $R$ is the rectangle
$[2,\,5]\,\times\,[1,\, 3]$ in the $xy$plane.
Solution: The integral is equal to the volume of
the solid $W$ shown to the right, namely the region between the
$x$$y$ plane and the plane $z=\frac{4}{3}(5x)$ and above $R$. To
find its volume, take a vertical slice for fixed $y, \, 1 \le y \le
3$. The trace of the solid on the vertical plane is the same triangle
for each $y$. But the triangle has height $=4$ and base $=3$, so it
has area $$\frac{1}{2}\times \hbox{base}\times
\hbox{height} \ = \ 6\,.$$
Thus $W$ has
$$\hbox{volume} = \hbox{area triangle}\times \hbox{sidelength} = 12,
\qquad \hbox{and}$$
$$\iint_R \frac{4}{3}(5x) dA = \hbox{ volume of $W$ } = 12.$$

