Math · Calculus I · Concept

Riemann sums: left, right and midpoint

A Riemann sum approximates a definite integral by adding rectangle areas: split [a, b] into n pieces of width Δx = (b − a)/n, take a height f(xᵢ*) in each piece, and add the products f(xᵢ*)·Δx. Left, right and midpoint sums choose different heights; as n grows, all of them approach the integral.

Height times width

Split [a, b] into n equal pieces of width Δx. In each piece choose a sample point xᵢ*, multiply the height f(xᵢ*) by Δx, and add. Each product is the signed area of one rectangle.

S_{n} = \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x, Δ x = \frac{b - a}{n}

\begin{array}{c} S_{n} = \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x \\ Δ x = \frac{b - a}{n} \end{array}

Left, right and midpoint sums

The name says where each rectangle takes its height. For a function that increases on the interval, the left sum is too small and the right sum too big; for a decreasing function it is the other way round. The midpoint sum is usually much closer than either.

Where each sum takes its height on [xᵢ₋₁, xᵢ]
Sum	Height	If f is increasing
Left, Lₙ	f(xᵢ₋₁)	Underestimates
Right, Rₙ	f(xᵢ)	Overestimates
Midpoint, Mₙ	f at the midpoint	Usually closest

Sigma notation

Σ means add: the index i runs from 1 to n, and each value of i contributes one term. The left sum uses the points x₀ to xₙ₋₁ and the right sum x₁ to xₙ, n heights in each case.

\sum_{i = 1}^{n} a_{i} = a_{1} + a_{2} + \dots + a_{n}

From sums to the integral

As n grows, the rectangles get thinner and, for a continuous function, every way of choosing the heights approaches the same number. That limit is the definition of the definite integral. A few agreeing sums are evidence, not proof: a finite sum can miss a narrow spike.

\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x

\begin{array}{c} \int_{a}^{b} f (x) d x \\ = lim_{n \to \infty} \sum_{i = 1}^{n} f (x_{i}^{*}) Δ x \end{array}

Signed area

Where f is negative, the rectangles have negative height and subtract. So a Riemann sum, like the integral, measures signed area. Total geometric area uses |f| instead.

Trapezoids average left and right

Joining neighboring heights with straight segments gives trapezoids. The trapezoidal sum is the average of the left and right sums.

T_{n} = \sum_{i = 1}^{n} \frac{f (x_{i - 1}) + f (x_{i})}{2} Δ x = \frac{L_{n} + R_{n}}{2}

\begin{array}{c} T_{n} = \sum_{i = 1}^{n} \frac{f (x_{i - 1}) + f (x_{i})}{2} Δ x \\ = \frac{L_{n} + R_{n}}{2} \end{array}

Common mistakes

Using n + 1 heights: n rectangles need n heights. The left sum stops at xₙ₋₁ and the right sum starts at x₁.
Forgetting to multiply by Δx, or using b − a as the width instead of (b − a)/n.
Treating signed area as total area: rectangles below the axis subtract.
Assuming the left sum is always too small: that holds only where f is increasing.
Expecting every increase in n to shrink the error: for a smooth function it shrinks overall, but a coarse grid can hit or miss a narrow feature by chance.

Key terms

Riemann sum: An approximation of the area under a curve: add up rectangles, each with a width Δx and a height from the function. Taking heights at left ends, right ends or midpoints gives different sums.
Sigma notation: A compact way to write a sum: Σ from i = 1 to n of aᵢ means a₁ + a₂ + ⋯ + aₙ. The index i takes each integer value from the lower limit to the upper limit, and each value contributes one term.
Interval partition: Splitting an interval into smaller pieces side by side. The pieces don’t have to be equal, but equal widths make Riemann sums easier.
Midpoint rule: Approximating an integral with rectangles whose heights are the function’s values at the midpoints of the subintervals. It is often more accurate than using left or right endpoints.
Definite integral: ∫ₐᵇ f(x) dx: the signed area between the graph of f and the x-axis from a to b, defined as the limit of Riemann sums. Area below the axis counts as negative.
Signed area: Area counted with a sign: regions above the x-axis count as positive and regions below as negative, so they can cancel. Total area counts every region as positive.

Work through an example

Estimate ∫₀¹ x² dx with n = 2 rectangles, using left, right and midpoint heights. Compare each estimate with the exact value 1/3.

Left, right and midpoint sums for x² →

Estimate distance from a table of speeds →

Sources and scope

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Open the example inputs, change a value and keep a useful result on your board.

Open Riemann Sums Check the sums in Math Open worked example on a board Riemann sum in Math Reference

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