Average Value of a Function

The average value of a function over a closed interval equals the height of a rectangle whose area matches the net area under the curve. This net area accounts for both positive and negative contributions of the function. For a positive function, this geometric interpretation provides an intuitive way to visualize how a single value summarizes the function's overall behavior across the interval.

The interval is divided into many small subintervals of equal width. On each subinterval, a representative function value is chosen as the height of a thin rectangle. Adding the areas of all rectangles estimates the total area under the curve. As the number of subintervals increases, this approximation becomes increasingly accurate and converges to the exact value through a limiting process.

Temperature varies continuously throughout a 24-hour period, but a single representative value is often more useful than tracking every fluctuation. The average temperature summarizes this continuous variation into one number that reflects the overall thermal behavior. This demonstrates how average function values simplify real-world data while preserving meaningful information about the system's behavior.

The limiting process of rectangular approximation leads to an expression involving a definite integral. The definite integral represents the exact accumulated value of the function over the interval. The average value equals this integral divided by the width of the interval, providing a precise mathematical formula that replaces the approximation method.

Net area accounts for both positive and negative contributions of the function across the interval. When a function dips below the x-axis, those negative regions subtract from the total. The average value uses net area, not total area, ensuring that the representative value accurately reflects the function's overall behavior including regions where it takes negative values.

As the number of subintervals increases and their widths decrease, the sum of rectangular areas becomes increasingly accurate. The approximation converges to the exact accumulated value of the function. This limiting process is fundamental to calculus and connects the discrete rectangular method to the continuous definite integral used in the average value formula.

Any continuously varying quantity—such as velocity, concentration, or pressure—can be summarized by its average value over a time or space interval. The method divides the interval into subintervals, estimates contributions, and finds the representative value. This approach works for area between curves problem solving and other applications of integration across diverse scientific and engineering contexts.