Riemann sums and the midpoint rule

With just four rectangles, the midpoint sum for x² on [0, 2] is 2.625, already within 0.042 of the exact 8/3.

Any point in each piece

Left and right ends are two choices among many. In general, pick any point x_i* in the i-th piece and use f(x_i*) as the height. The total Σ_i=1ⁿ f(x_i*)Δx is called a Riemann sum, named, as OpenStax notes, after the 19th-century mathematician Bernhard Riemann, who developed the idea. Left and right sums are Riemann sums; so is the midpoint sum M_n, which takes each height at the middle of its piece.

The midpoint sum is usually much better than either endpoint sum. For x² on [0, 2] with n = 4, the midpoints are 0.25, 0.75, 1.25 and 1.75. Their squares are 0.0625, 0.5625, 1.5625 and 3.0625, which add to 5.25; times Δx = 0.5 gives M₄ = 2.625. Compare: L₄ = 1.75 and R₄ = 3.75 were off by about 0.92 and 1.08. The reason is visible on a graph. In each piece, the rectangle cuts off a small sliver of the curve on one side and adds a similar sliver on the other, and the two nearly cancel.

The table above, rounded to four decimal places, shows the pattern for x² on [0, 2]. Each time n doubles, the endpoint errors roughly halve, while the midpoint error drops to about a quarter: 0.042, 0.010, 0.0026, 0.00065. All three columns head for the same number, 2.6666…, and the definite integral in the next unit is defined as exactly that common limit.

There is a quick link between the three sums. The average of the left and right sums, (L_n + R_n)/2, is the trapezoid rule, which replaces each rectangle with a trapezoid joining the two ends of the curve. For x² on [0, 2] with n = 4 it gives (1.75 + 3.75)/2 = 2.75, too high by about 0.083, roughly twice the midpoint error and on the other side. That pattern, trapezoids overshooting a curve that bends upward while midpoint rectangles undershoot it by about half as much, holds for many smooth curves and is the starting point for Simpson's rule in later courses.

Computers rely on exactly these sums. When a function has no convenient antiderivative, or when it is known only from measured data, a program adds thousands of midpoint or trapezoid pieces. The lab below caps n at 64 so you can still see the rectangles, but the pattern in the table continues: each doubling makes the midpoint error about four times smaller.

Upper and lower sums

If you want a guaranteed overestimate, choose x_i* where f is largest on each piece; the result is an upper sum. Choosing where f is smallest gives a lower sum, a guaranteed underestimate. The true area is always between them. For a function that only increases or only decreases on the interval, the largest and smallest values sit at the endpoints, so the upper and lower sums are just the right and left sums in some order. OpenStax's curve (x − 1)³ + 4 on [0, 2] is increasing, with L₄ = 7.5 and R₄ = 8.5, and both approach 8 as n grows.