The average of the finitely many \(mn\) values \(f(x_{ij}^*, y_{ij}^*)\) that we take in a double Riemann sum is given by
\begin{equation*}
\mbox{Avg} _{mn} = \frac{1}{mn} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*).
\end{equation*}
If we take the limit as \(m\) and \(n\) go to infinity, we obtain what we define as the average value of \(f\) over the region \(R\text{,}\) which is connected to the value of the double integral. First, to view \(\text{ Avg } _{mn}\) as a double Riemann sum, note that
\begin{equation*}
\Delta x = \frac{b-a}{m} \ \ \ \ \ \text{ and } \ \ \ \ \ \Delta y = \frac{d-c}{n}.
\end{equation*}
Thus,
\begin{equation*}
\frac{1}{mn} = \frac{\Delta x \cdot \Delta y}{(b-a)(d-c)} = \frac{\Delta A}{A(R)},
\end{equation*}
where \(A(R)\) denotes the area of the rectangle \(R\text{.}\) Then, the average value of the function \(f\) over \(R\text{,}\) \(f_{\operatorname{AVG}(R)}\text{,}\) is given by
\begin{align*}
f_{\operatorname{AVG}(R)} \amp = \lim_{m,n \to \infty} \frac{1}{mn} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*)\\
\amp = \lim_{m,n \to \infty} \frac{1}{A(R)} \sum_{j=1}^n \sum_{i=1}^m f(x_{ij}^*, y_{ij}^*) \cdot \Delta A\\
\amp = \frac{1}{A(R)} \iint_R f(x,y) \, dA.
\end{align*}
Therefore, the double integral of \(f\) over \(R\) divided by the area of \(R\) gives us the average value of the function \(f\) on \(R\text{.}\) Finally, if \(f(x, y) \geq 0\) on \(R\text{,}\) we can interpret this average value of \(f\) on \(R\) as the height of the box with base \(R\) that has the same volume as the volume of the surface defined by \(f\) over \(R\text{.}\)