Riemann Integral

Key idea

Defined by upper and lower Darboux sums.

In depth

Define upper and lower sums using the supremum and infimum of f on each subinterval of a partition. f is Riemann integrable if the infimum of upper sums equals the supremum of lower sums, and that value is the integral.

Worked example

For f(x) = x on [0, 1], upper and lower sums with n pieces differ by 1/n, squeezing to 1/2.

Real-world uses

• Defining exact areas and totals for measured quantities such as flow or rainfall

Common mistake

Thinking all bounded functions are integrable.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.