Riemann Sums

Summary

A Riemann sum approximates a definite integral by summing sample heights times subinterval widths. Refining the partition makes the sum approach the integral when $f$ is Riemann integrable.

Prerequisites

Limits, Integrals

Definition

Partition $[a,b]$ into $n$ subintervals of width $\mathrm{\Delta }x=(b-a)/n$ (equal partitions for simplicity). Choose sample points $x_i^{\ast }$ in the $i$ -th subinterval. The Riemann sum is

Common choices: left endpoints, right endpoints, or midpoints.

For a rectangle $[a,b]\times [c,d]$ and $f(x,y)$ ,

Conditions / Assumptions

• Continuous functions on compact rectangles are Riemann integrable; the mesh of the partition must tend to zero.
• State whether left, right, or midpoint samples are used—do not mix them inconsistently.

Worked Example

Approximate ${\int }_0^1{x}^{2}dx$ with $n=4$ midpoint rule: $\mathrm{\Delta }x=1/4$ , midpoints $1/8,3/8,5/8,7/8$ .

The exact value is $1/3\approx 0.333\dots$ .

Double integral midpoint sample for ${\iint }_{[0,1{]}^2}xydA$ with a $2\times 2$ grid: midpoints $(1/4,1/4)$ , $(3/4,1/4)$ , $(1/4,3/4)$ , $(3/4,3/4)$ , and $\mathrm{\Delta }x\mathrm{\Delta }y=(1/2)(1/2)=1/4$ .

which matches ${\int }_0^1{\int }_0^{1}xydxdy=1/4$ exactly for this function and grid.

Common Mistakes

• Using wrong sample points (e.g. calling endpoints “midpoints”).
• Multiplying by $\mathrm{\Delta }x\mathrm{\Delta }y$ incorrectly (using $0.25\cdot 0.25$ when each side step is $0.5$ ).

Connections

• Limit of Riemann sums: definition of Integrals
• Multivariable: Double Integrals, Fubini's Theorem

References

Riemann sums are introduced in OpenStax Calculus Volume 1.^[1]