Name: estimating snowfall with midpoint rule
Uploaded: 2026-06-08T09:38:56.699Z
Duration: 1 min 27 s
Channel: JoVE

Imagine measuring snowfall in centimeters across Colorado over a 2-day period, with different regions receiving varying amounts. A contour map represents this variation across the state, showing lines of equal snowfall.To find the average snow depth, we need to calculate the average value of the variable snowfall function over a rectangular region.The total snowfall over the region can be expressed as a double integral. To estimate this integral using the midpoint rule, the region is divided into smaller equal sub-rectangles. Each sub-rectangle represents an equal portion of the total area.Within each sub-rectangle, a midpoint is identified. These points serve as representative locations for each section.The snowfall is evaluated at each midpoint, providing an approximate value for each subregion. For each sub-rectangle, a contribution is obtained by multiplying its area by the corresponding snowfall value.The contributions from all sub-rectangles are then summed to obtain an approximation of the total snowfall over the region. Finally, dividing this total by the area of the region gives the average snowfall.

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 RegionThe 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.

Average Value on a Rectangle

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estimating snowfall with midpoint rule

Estimating snowfall with the midpoint rule uses a rectangular region and a snow depth function of two variables. A snowfall map can show how snow depth changes across an area, such as Colorado, over a fixed time period. If f(x,y) gives the snow depth at a point in the region, then the average snowfall is found by comparing the total snowfall with the area of the region.

The total snowfall over the rectangle can be written as a double integral. This integral adds the snowfall from every point in the region. When the snowfall data comes from a contour map, finding the exact integral may be difficult. In that case, a numerical method such as the midpoint rule gives a useful approximation.

To use the midpoint rule, the rectangular region is split into smaller equal sub-rectangles. Each sub-rectangle has the same area, so each one represents an equal part of the total region. A midpoint is chosen in each sub-rectangle, and the snowfall function is evaluated at that point. That midpoint value stands in for the snowfall depth across the whole sub-region.

For each sub-rectangle, the approximate snowfall contribution is found by multiplying the sub-rectangle area by the midpoint value. Adding all of these contributions gives an estimate 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) is the total area of the rectangular domain. This midpoint approximation turns changing snow depth across a region into one estimated average value.

Estimating Snowfall with Midpoint Rule

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Estimating snowfall with the midpoint rule uses a rectangular region and a snow depth function of two variables. A snowfall map can show how snow depth changes across an area, such as Colorado, over a fixed time period. If f(x,y) gives the snow depth at a point in the region, then the average snowfall is found by comparing the total snowfall with the area of the region. The total snowfall over the rectangle can be written as a double integral. This integral adds the snowfall from every point...

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Learn how the midpoint rule and double integrals estimate total snowfall and average snow depth over a rectangular region from a contour map.

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The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

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Estimating Snowfall with Midpoint Rule

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