Average Value on a Rectangle

A snowfall map can represent how snow depth varies across a region, such as Colorado, over a fixed time period. Since different locations may receive different amounts of snow, the snowfall depth is described by a function of two variables. If f(x,y) represents the snow depth at a point in a rectangular region, then the average snowfall over the entire region is found by comparing the total accumulated snowfall with the area being measured.

Total Snowfall over a Region

The total snowfall over a rectangular region can be represented by a double integral. This integral adds the snowfall contributions from every point in the region. Since evaluating the exact integral may be difficult when the snowfall data comes from a contour map, a numerical method such as the midpoint rule can be used to approximate it.

To apply the midpoint rule, the rectangular region is divided into smaller equal sub-rectangles. Each sub-rectangle represents an equal portion of the total area. A midpoint is then identified within each sub-rectangle, and the snowfall function is evaluated at that midpoint. This midpoint value is used as a representative snowfall depth for the entire sub-region.

For each sub-rectangle, the approximate snowfall contribution is found by multiplying the area of the sub-rectangle by the snowfall value at its midpoint. Adding all of these contributions gives an approximation of the total snowfall over the region:

\begin{equation*}\iint_R f(x,y)\,dA \approx \sum f(x_i^*,y_i^*)\,\Delta A \end{equation*}

The average snowfall is then found by dividing the total snowfall by the area of the rectangular region:

\begin{equation*}\text{Avg Snowfall} \approx \frac{1}{A(R)} \sum f(x_i, y_i)\,\Delta A\end{equation*}

Here, A(R) represents the total area of the rectangular domain.

Using the midpoint approximation, this becomes an estimated average based on representative points from each sub-rectangle. This process converts regional snowfall variation into a single average value that describes the overall snow depth across the region.