---
title: "Riemann Sums and the Definite Integral as a Limit"
description: "Build the definite integral as the limit of sums of rectangle areas, with sample points anywhere in each subinterval. This is the definition every later property of the integral is argued from."
canonical: https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9
source: https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9.md
retrieved: 2026-09-12
---

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# Riemann Sums and the Definite Integral as a Limit

Build the definite integral as the limit of sums of rectangle areas, with sample points anywhere in each subinterval. This is the definition every later property of the integral is argued from.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/riemann-sums-and-the-definite-integral-as-a-limit-mt_pwa2SRU6P9

## Ready when they can

- Write a Riemann sum with n subintervals in sigma notation for a given function
- Compute left, right and midpoint sums for the same integral and compare them
- Explain how refining the partition removes the dependence on where samples are taken

## Lesson: Areas from thinner and thinner rectangles

A Riemann sum estimates area with rectangles. Split a to b into n equal pieces of width b minus a over n, pick a sample point in each piece, and add height times width, written compactly with sigma notation.

**Example.** Left sums take heights at each left edge, right sums at each right edge, and midpoint sums at each center. For x squared on 0 to 2 with 4 pieces of width 0.5, the left sum is 1.75, the midpoint sum is 2.625, and the right sum is 3.75. On a climbing function the left sum lands low and the right sum lands high.

The definite integral is the limit of these sums as n grows without bound and widths shrink toward 0. It gives net area: parts above the axis count positive and parts below count negative. A continuous function is always integrable, so the limit exists.

**Tip.** Refining the partition washes out where you sampled. Thin pieces leave heights almost identical across each piece, so left, right, and midpoint sums squeeze to one shared limit regardless of sample spots.

**Recap.** Slice into rectangles, add them with sigma notation, then push the count to infinity for the exact integral.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [The Trapezium Rule](https://lightmysky.com/learn/mathematics/the-trapezium-rule-mt_rAlJ1WIRmZ)
- [The Definite Integral and the Area Under a Curve](https://lightmysky.com/learn/mathematics/the-definite-integral-and-the-area-under-a-curve-mt_S2IO2S7kgd)

## Opens up

- [The Riemann Integral and When a Function Is Integrable](https://lightmysky.com/learn/mathematics/the-riemann-integral-and-when-a-function-is-integrable-mt_Ahl2MYIA4N)
- [The Fundamental Theorem of Calculus](https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A)
- [The Lebesgue Integral and What It Repairs](https://lightmysky.com/learn/mathematics/the-lebesgue-integral-and-what-it-repairs-mt_KO92nLc5YU)
- [Double Integrals over General Regions](https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw)