What Is a Riemann Sum?

A Riemann sum is not the area under a curve. It is an approximation of that area, a finite sum of rectangles that gets closer to the true value only as the number of rectangles increases without limit. The question "what is a riemann sum" is answered most clearly by looking at a picture: imagine the curve y = f(x) on the interval [a, b]. Below it, draw vertical rectangles that touch the curve at one point each. Their total area is the Riemann sum. The definition from Stewart's Calculus (Section 5.1, The Area Problem) starts exactly here: break the region, sum the rectangles, then take the limit. That limit is the definite integral.

The Riemann sum itself is always a number you can compute by hand or with a calculator. The integral is the number you get after an infinite process. Confusing the two is the most common error a student makes on the first encounter.

Draw the curve f(x) = x² from x = 0 to x = 2. Below it, draw four rectangles of equal width. Each rectangle's height touches the curve at its left edge. The sum of those four rectangle areas is a left Riemann sum. The rectangles leave noticeable gaps between their tops and the curve, gaps that shrink as you draw more rectangles. OpenStax Calculus Volume 1 (Section 5.1, Approximating Areas) shows this exact diagram: a curve, a set of rectangles, and the caption "approximating area with rectangles." The picture makes the concept immediate. The rectangles do not fill the curve exactly. The sum is not the area. It is an estimate that improves when you increase the number of rectangles.

Every Riemann sum has four parts. The partition is the set of points that divide [a, b] into subintervals. For a uniform partition, Δx = (b − a)/n, where n is the number of subintervals. The sample point is the x-value inside each subinterval where you evaluate the function to get the height. The width Δx is constant for uniform partitions but can vary for non-uniform ones. Stewart's notation (Section 5.2) labels the partition points as x₀ = a, x₁, x₂, ..., xₙ = b. The sample point in the i‑th subinterval is written xᵢ*. The height is f(xᵢ*). The area of one rectangle is f(xᵢ*) Δx. The Riemann sum is the sum of those rectangle areas: Σ_{i=1}^{n} f(xᵢ*) Δx.

A left Riemann sum uses the left endpoint of each subinterval as the sample point; for an increasing function it underestimates the area. A right Riemann sum uses the right endpoint; for an increasing function it overestimates. A midpoint Riemann sum uses the midpoint of each subinterval; it is often more accurate than either left or right for the same n, and its over‑ or underestimate depends on concavity, not monotonicity.

Approximate the area under f(x) = x² from x = 0 to x = 2 using a left Riemann sum with n = 4 subintervals.

Δx = (2 − 0) / 4 = 0.5. The left endpoints are x = 0, 0.5, 1.0, 1.5. Evaluate f at each: f(0) = 0, f(0.5) = 0.25, f(1) = 1, f(1.5) = 2.25. Sum the heights: 0 + 0.25 + 1 + 2.25 = 3.5. Multiply by Δx: 3.5 × 0.5 = 1.75.

The exact integral of x² from 0 to 2 is 8/3. The left sum 1.75 underestimates because the function is increasing, each left‑endpoint height is lower than the curve over most of the subinterval. Increasing n to 8 reduces the error. The AP Calculus CED (Topic 6.2) requires you to justify this over‑ or underestimate by stating that the function is increasing or decreasing on the interval.

The definite integral is defined as the limit of Riemann sums as the number of subintervals goes to infinity and the maximum subinterval width (the mesh) goes to zero. Stewart (Section 5.2) gives the definition: ∫_a^b f(x) dx = lim_{n→∞} Σ_{i=1}^{n} f(xᵢ*) Δx. This limit is the exact signed area. The Riemann sum is the finite, computable step before the limit. The limit of a Riemann sum is a separate topic, it appears on the limit of riemann sum page, but the key idea is that no finite sum, no matter how large n is, equals the integral. Only the limit does.

Students bring three persistent misunderstandings to Riemann sums. Each one can lose points on an exam or produce wrong answers in applied work.

"A Riemann Sum Gives The Exact Area"

This is the most damaging misconception. A Riemann sum is an approximation. The exact area is the definite integral, which is the limit of Riemann sums. Treating a finite sum as the true value misses the entire point of the definition. The AP scoring guidelines penalise any answer that calls a left or right sum "the area" without qualification.

"More Rectangles Always Means A Better Approximation"

For a fixed function and a fixed rule (left, right, or midpoint), increasing n generally reduces the error, but not monotonically. A larger n can occasionally give a worse approximation for a specific function. The error bound formulas, Stewart (Section 7.7) gives |E_L| ≤ (b−a)²/(2n) × max|f'|, guarantee that the error eventually shrinks as n increases, but not that every step improves the estimate.

"Negative Area Is Not Area"

The definite integral gives signed area: area above the x‑axis is positive, area below is negative. A Riemann sum that includes subintervals where f(x) is negative will produce negative terms. Students who treat all areas as positive get the wrong sum. The AP Calculus CED (Topic 6.3) defines the integral as signed area, and free‑response questions often include a part where the function dips below the x‑axis.