Concavity and the Direction of Trapezoidal Error

Why Concavity Predicts the Direction of Error

A trapezoidal estimate replaces part of a function’s graph with a straight line segment joining its endpoint values. If the graph bends upward, that segment tends to lie above the graph; if it bends downward, the segment tends to lie below it. Since the area of a trapezoid is the integral of its straight-line top, this comparison predicts whether the estimate is greater or less than the signed integral.

The prediction is useful before doing any arithmetic: it provides a way to check whether an estimate is plausible. The comparison concerns signed integrals, so it remains meaningful even when a graph lies below the horizontal axis.

The Trapezoidal Estimate and Its Sign

For one interval \([a,b]\), the trapezoidal estimate uses the line through \((a,f(a))\) and \((b,f(b))\). The width is \(b-a\), and the average of the two endpoint heights is \(\frac{f(a)+f(b)}{2}\). Thus the estimate is

For a partition \(a=x_0<x_1<\cdots<x_n=b\), the composite trapezoidal estimate adds the estimate on each subinterval:

Here, “overestimate” means \(T\ge \int_a^b f(x)\,dx\), and “underestimate” means \(T\le \int_a^b f(x)\,dx\). These comparisons refer to signed quantities, not necessarily to ordinary geometric area.

Why the Chord Lies on the Predicted Side

Let \(L(x)\) be the line through the endpoints of the graph on \([a,b]\). Its equation is

Suppose \(f''\ge 0\). The mean value theorem applied to \(f'\) shows that \(f'\) is nondecreasing: for \(u<v\), some \(c\in(u,v)\) satisfies \(f'(v)-f'(u)=f''(c)(v-u)\ge 0\). Now choose any \(x\) with \(a<x<b\). Applying the mean value theorem to \(f\) on \([a,x]\) and \([x,b]\) gives points \(u\in(a,x)\) and \(v\in(x,b)\) such that

Because \(u<v\) and \(f'\) is nondecreasing, the first slope is no greater than the second. Multiplying by the positive number \((x-a)(b-x)\) and rearranging gives

Thus, when \(f\) is concave up, its graph is at or below the chord. Integrating \(f(x)\le L(x)\) over \([a,b]\) preserves the inequality. The integral of \(L\) is the trapezoidal estimate, so \(\int_a^b f(x)\,dx\le T\). If \(f''\le 0\), the slope comparisons reverse, giving \(f(x)\ge L(x)\) and therefore \(\int_a^b f(x)\,dx\ge T\). This proves the stated direction of error.

Worked Examples: Testing the Prediction

Common Mistake: Reversing the Chord Comparison

A common error is to think that a concave-up graph lies above the line segment joining its endpoints. That reverses the relationship. For example, on \([0,1]\), \(f(x)=x^2\) is concave up and its endpoint chord is \(L(x)=x\). At \(x=\frac12\),

The chord is above the graph, not below it. The trapezoidal estimate is \(T=\frac{1}{2}(0+1)=\frac12\), while the exact integral is \(\int_0^1 x^2\,dx=\frac13\). Thus \(T>\frac13\), confirming that the concave-up case is an overestimate.

When the Prediction Needs Care

The direction-of-error theorem requires the stated concavity condition over the interval being estimated. If \(f''\) changes sign, the graph may be below its chord on some parts and above it on others. Their contributions can compete, so concavity alone may not determine the direction of the total error. Check the second derivative on the entire interval, or on each subinterval of a composite estimate, before applying the prediction.

If \(f''\) is zero throughout an interval, then \(f\) is linear there and its chord matches the graph; the trapezoidal estimate is exact. More generally, non-strict concavity gives a non-strict comparison: equality is possible, while strict concavity throughout the interior gives a strict inequality.

The practical value of this result is that it connects a local visual feature—the direction of bending—to a reliable comparison of numerical and exact integrals. Compute \(f''\), determine its sign on the interval, predict the direction, and then use exact values when available to check the estimate.