Average Value of a Function

The average value of a function over a closed interval can be interpreted geometrically as the height of a rectangle whose area equals the net area under the curve across that interval. This net area accounts for both positive and negative contributions of the function, providing a single representative value that reflects the function’s overall behavior

A practical illustration of this idea arises when monitoring the temperature inside a greenhouse over a twenty-four-hour period. Although the temperature varies continuously throughout the day, it is often useful to summarize this variation with a single number that represents the average temperature over the entire time interval.

To find out this average, the interval is divided into many small subintervals of equal width. On each subinterval, a representative value of the function is chosen and used as the height of a thin rectangle. The area of each rectangle approximates the contribution of the function over that short segment. Adding the areas of all such rectangles provides an estimate of the total area under the curve.

As the number of subintervals increases and their widths decrease, this approximation becomes increasingly accurate. In the limit, the sum of the rectangular areas converges to the exact accumulated value of the function over the interval. This limiting process corresponds to the definite integral shown above.

When the function remains positive over the interval, the average value can be visualized as the height of a rectangle whose area matches the area under the curve. This geometric interpretation provides a clear and intuitive understanding of how the average value summarizes the behavior of a continuously varying quantity.