Function Averages Versus Endpoint Averages

Two Different Ways to Average a Function

An average can summarize many values with a single number. For a function on an interval, one natural summary uses all the function’s values across the interval. Another uses only the values at the two endpoints. These summaries sound similar, but they need not be equal.

The distinction matters whenever the function describes a quantity that varies across time or space. The integral average accounts for the entire interval, while the endpoint average ignores what happens between its ends. Comparing them helps identify when endpoint information is sufficient and when it is not.

The factor \(1/(b-a)\) in the integral average divides the total accumulated value by the interval’s length. Thus, if \(f\) has units such as liters per minute, both averages have those same units. The endpoint average has the same units because it averages two values of \(f\).

These definitions use different information. The integral average depends on the values throughout \([a,b]\). The endpoint average depends only on \(f(a)\) and \(f(b)\). To see when they coincide—and why they often do not—we first examine a case where the function changes at a constant rate.

Why Linear Functions Give the Same Average

A function is affine (often called linear in this context) if it has the form \(f(x)=mx+c\), where \(m\) and \(c\) are constants. Its values change uniformly between the endpoints. For every such function, the integral average and endpoint average are equal.

Here is a direct derivation. An antiderivative of \(mx+c\) is \(\frac{m}{2}x^2+cx\), so

At the endpoints, \(f(a)=ma+c\) and \(f(b)=mb+c\). Their arithmetic mean is

The two expressions are identical, proving the claim. The result reflects the uniform change in an affine function: its values throughout the interval are distributed evenly between its endpoint values.

A Line Through the Endpoint Values

For any continuous function, not just an affine one, there is an affine function that agrees with it at both endpoints. Call this function \(L\):

This formula gives \(L(a)=f(a)\) and \(L(b)=f(b)\). Its integral average is the endpoint average of \(f\). Indeed, integrating the displayed formula gives

Dividing by \(b-a\), the integral average of \(L\) is \((f(a)+f(b))/2\). This gives a useful exact comparison for \(f\):

This identity explains why the endpoint average can fail to capture the function average: the endpoint values determine \(L\), but they do not determine how \(f\) behaves relative to \(L\) between the endpoints. If \(f-L\) has positive and negative parts, their contributions to the integral can cancel. Consequently, the averages can agree even when the function is not a line.

When the Averages Differ

The difference can go the other way. For example, consider a function whose values between the endpoints rise above the line joining them.

Together, these examples show that there is no universal rule that the endpoint average is above or below the integral average. The direction depends on how the function lies relative to the line joining its endpoint values.

Agreement Does Not Require a Linear Function

The affine-function theorem gives a sufficient condition for agreement, not a necessary one. A nonlinear function can have positive and negative deviations from its endpoint line whose signed integral totals zero.

In terms of the comparison identity, agreement means \(\int_a^b(f(x)-L(x))\,dx=0\). It does not require \(f(x)-L(x)=0\) at every point. Positive and negative deviations may balance exactly.

A Common Mistake

The endpoint calculation is not wrong; calling it the function’s integral average is the error. The two quantities agree for affine functions, but for a general function the intervening values must be accounted for by the integral. A useful check is to ask whether the function is known to be affine or whether its behavior between the endpoints has been evaluated.

How to Compare the Averages

For a continuous function on \([a,b]\), a careful comparison can be organized as follows:

The endpoint average is a compact summary of two values. The integral average is a summary of the full function over an interval. The line joining the endpoint values provides the bridge between them: the endpoint average is the integral average of that line, and the difference for the original function is exactly its average signed deviation from the line.