Why Average Value Matches a Rectangle Height

Why a Rectangle Can Represent an Average

A function may vary throughout an interval, so its graph does not have one constant height. Yet its total signed accumulation over that interval can be matched by a rectangle: use the same base, and choose one constant height that gives the same signed area.

That constant height is the function’s average value on the interval. The idea is not that the function stays at this height, or even that it must take this value. Rather, the constant rectangle and the varying function have the same net accumulation across the same horizontal distance.

Signed Accumulation and Rectangle Area

Let \(f\) be integrable on an interval \([a,b]\), where \(a<b\). The definite integral \(\int_a^b f(x)\,dx\) represents signed accumulation: contributions above the horizontal axis are positive, and contributions below it are negative. If \(f\) describes a rate, this integral is the net change accumulated over the interval.

Now consider a constant function with value \(c\). Its graph is a horizontal line, so the signed area between its graph and the horizontal axis over \([a,b]\) is a rectangle. Its base has length \(b-a\), and its signed height is \(c\). Therefore its signed area is \((b-a)c\). When \(c<0\), the rectangle’s contribution is negative.

The word signed matters. A rectangle below the axis does not have a negative geometric size, but its signed area in an integral is negative. This sign records the direction of accumulation.

Deriving the Average-Value Formula

To find the constant height that represents \(f\), require the constant rectangle to have exactly the same signed accumulation as the function. In integral form, the requirement is that the integral of the constant function \(c\) equal the integral of \(f\) over the same interval.

In equations, the matching condition and its solution are:

This derivation also explains the conditions in the formula. The definite integral must exist, so \(f\) must be integrable on the interval; continuity on \([a,b]\) is a standard sufficient condition. The interval must have positive length so division by \(b-a\) is possible. If \(a=b\), the base has zero length and this definition does not give an average value.

Why the Matching Height Is Unique

There cannot be two different constant heights that give the same signed rectangle area over an interval of positive length. If heights \(c_1\) and \(c_2\) both match the function, then

Subtracting the two sides that contain the heights gives \((c_1-c_2)(b-a)=0\). Because \(b-a>0\), it follows that \(c_1-c_2=0\), so \(c_1=c_2\). Thus the matching signed area determines exactly one rectangle height.

The argument does not require \(f\) to be positive. It only requires its signed integral to exist. If the net signed accumulation is negative, the matching constant height is negative as well.

Worked Example: A Triangle Above the Axis

Worked Example: Positive and Negative Regions

Worked Example: A Rate That Changes Sign

Common Pitfall: Using Total Area Instead of Signed Area

When a graph crosses the axis, adding the geometric areas without signs gives total area, not the signed accumulation used to define average value. Returning to the two-triangle example, the geometric areas are \(6\) and \(3\). A mistaken calculation would add them and set \(6c=6+3\), giving \(c=\frac{3}{2}\).

That answer is incorrect for a signed-accumulation-matching rectangle. The lower triangle contributes \(-6\), not \(+6\), so the correct equation is

Checking the result makes the distinction clear: the wrong height \(\frac{3}{2}\) gives rectangle area \(6(\frac{3}{2})=9\), not the graph’s signed area of \(-3\). The correct height gives \(6(-\frac{1}{2})=-3\).

What the Average Height Does—and Does Not—Say

The average height compresses the function’s net accumulation into one constant value over a specified interval. It depends both on the function and on the chosen interval: changing the interval changes the rectangle’s base and may change the signed accumulation being matched.

It is also important to distinguish the average height from the function’s actual values. The definition guarantees a constant rectangle with matching signed area. By itself, it does not say that the graph reaches that height at some input. The rectangle is an accumulation comparison, not a claim that the varying graph is flat or passes through a particular point.

For a rate function, the interpretation is especially useful. The integral gives net change, and dividing that net change by elapsed time gives the constant rate that would produce the same net change over that time. A negative matching rate indicates negative net change over the interval; it does not mean that every instantaneous rate was negative.