Concavity and Midpoint Over- or Underestimates

Why Curvature Predicts Midpoint Error

A midpoint sum uses the function value at the center of each subinterval to set the height of a rectangle. If the graph bends upward or downward, that central height can be consistently above or below the function’s average height across the subinterval. Concavity tells us which direction to expect.

The rule is not about whether a function is increasing or decreasing. It is about how the graph bends. A function that bends upward has a midpoint value below its average value on a symmetric interval, so its midpoint rectangle underestimates. A function that bends downward has a midpoint value above its average, so its midpoint rectangle overestimates.

Concavity and the Midpoint Sum

A function is concave up on an interval if its graph bends upward there, like a cup. It is concave down if its graph bends downward, like an arch. A standard way to determine concavity, when the function is twice differentiable, is to examine its second derivative: \(f''(x)\ge 0\) throughout an interval implies that \(f\) is concave up there, and \(f''(x)\le 0\) implies that it is concave down.

For a continuous function on \([a,b]\), the midpoint estimate using one subinterval has width \(h=b-a\) and midpoint \(m=(a+b)/2\). Its midpoint sum is the rectangle’s signed area, \(h f(m)\). The integral records the function’s signed accumulation over the interval.

If the interval is divided into \(n\) equal subintervals, each has width \(\Delta x=(b-a)/n\). Let \(x_i^*\) be the midpoint of the \(i\)-th subinterval. The composite midpoint sum is

When the same concavity holds throughout each subinterval, the direction of the error is determined separately on each one. Adding those inequalities gives the direction for the full midpoint sum.

Why the Midpoint Rule Works

The key idea is to compare the function value at the center with function values at pairs of points equally far to either side. For a concave-down function, the center value is at least the average of each such pair. This is precisely the defining inequality for concavity, applied to two points with equal weights.

Here is a derivation for one subinterval \([a,b]\). Let \(m=(a+b)/2\). For any \(t\) between \(0\) and \((b-a)/2\), the points \(m-t\) and \(m+t\) are equally spaced around \(m\). Concavity down gives

Integrate both sides with respect to \(t\) from \(0\) to \((b-a)/2\). The left side is constant, so its integral is \(\frac{b-a}{2}f(m)\). On the right, the two terms together cover the interval from \(a\) to \(b\): the substitution \(x=m-t\) covers the left half, and \(x=m+t\) covers the right half. Therefore,

Multiplying both sides by \(2\) proves that the one-subinterval midpoint estimate is at least the integral. For concavity up, the inequalities reverse at the pairwise comparison and throughout the derivation. This proves the midpoint rule’s direction rather than merely suggesting it from a sketch.

Worked Examples

Reading the Graph and Checking the Conditions

On a graph, imagine a vertical line through the midpoint of a subinterval. For a concave-up curve, points equally far to the left and right of that line have an average height at least as large as the height at the center. The midpoint rectangle is therefore too short relative to the function’s average height. For a concave-down curve, the central height is at least as large as the average of those paired heights, so the rectangle is too tall.

This reasoning depends on concavity across the subinterval, not just at its midpoint. If the graph changes concavity within an interval, the theorem does not by itself determine the direction of the total error there. On a partition, if some subintervals are concave up and others concave down, their errors may point in opposite directions. Without further information, the total midpoint sum may be either an overestimate or an underestimate.

The conclusion applies to signed integrals too. A function need not stay above the horizontal axis for the concavity comparison to hold: the pairwise inequalities still compare its values. What matters is that the function has the stated concavity throughout the interval.

Common Mistake: Confusing Concavity with Increase

A rising graph does not automatically make a midpoint sum an overestimate. For example, \(f(x)=x^2\) is increasing on \([0,2]\), but it is also concave up there. Its one-rectangle midpoint estimate is \(2\), while its integral is \(8/3\), so the estimate is an underestimate. The mistaken reasoning “the graph rises, so the midpoint rectangle must be too high” uses monotonicity where concavity is needed.

To avoid this mistake, check the second derivative or the graph’s bending on the whole subinterval. Increasing or decreasing describes the direction of slope; concave up or concave down describes how the slope changes. Only the latter gives this midpoint-error rule.

When the Estimate Is Exact

If a function is linear on a subinterval, its midpoint rectangle gives the exact integral there. The heights on either side of the midpoint balance around the midpoint height, so the average height equals the central height. This is the equality case in the concavity comparison. For a strictly concave-up or strictly concave-down function, the corresponding midpoint inequality is strict, and the estimate is not exact on an interval of positive width.