The contour map shows the temperature, in degrees Celsius,...

The contour map shows the temperature, in degrees Celsius, at $4: 00$ PM on a day in February in Colorado. (The state measures $624 \mathrm{~km}$ west to east and $444 \mathrm{~km}$ south to north.) Use the Midpoint Rule with $m=n=4$ to estimate the average temperature in Colorado at that time.

Key Concepts

Double Integrals

Double integrals extend the concept of single-variable integration to functions of two variables, allowing the calculation of volume under a surface over a given region. They are fundamental in applications that involve computing aggregate quantities like mass, charge, or, as in this context, average temperature over an area.

Riemann Sum

A Riemann sum approximates the value of an integral by partitioning the domain into small subregions and summing the products of the function values at chosen sample points and the areas of these subregions. This method is central to numerical integration techniques when an exact antiderivative is hard to obtain.

Midpoint Rule

The Midpoint Rule is a specific type of Riemann sum where the function is evaluated at the center (midpoint) of each subregion. This approach often yields a better approximation compared to other point choices, as it balances over- and under-estimation of the function's behavior over each subregion.

Partitioning of Regions

Partitioning a region involves dividing the domain into smaller, manageable subregions, typically rectangles in double integration contexts. This process is crucial for numerical integration methods as it lays the groundwork for applying techniques like the Riemann sum or the Midpoint Rule by ensuring a systematic approximation across the entire area.

Average Value of a Function

The average value of a function over a region is computed by integrating the function over that region and dividing by the total area of the region. This concept is key in situations where one needs a representative value from variations of intensity, such as temperature across a geographic area.

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