CalcRef

Integrals · Fundamental Theorems

Riemann Sum

∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗)Δx\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x

The definite integral is defined as the limit of Riemann sums. Partition [a,b] into n subintervals of width Δx = (b-a)/n.

  • 3 variables
  • 2 worked examples
  • 19 in Integrals

Variables

Variables used in the Riemann Sum formula
SymbolNameUnit
aLower bound -
bUpper bound -
nNumber of subintervals -

Worked examples

Approximate ∫₀¹ x² dx using a right Riemann sum with n = 4.
  1. Δx = 1/4. Right endpoints: 1/4, 1/2, 3/4, 1
  2. Sum = (1/16 + 1/4 + 9/16 + 1)(1/4) = (1/16 + 4/16 + 9/16 + 16/16)(1/4)
  3. = (30/16)(1/4) = 30/64 = 15/32

Answer: 15/32 = 0.46875 (exact: 1/3 ≈ 0.3333)

1 more worked examplePremium

Approximate ∫₀¹ x² dx using a left Riemann sum with n = 4.

Step-by-step solution locked.

Unlock the full working. CalcRef Premium opens the step-by-step solution above, and 141 worked solutions across the whole library - the intermediate lines, not just the answer. The formula statement, its conditions, its variables and the first worked example stay free on every page.
See PremiumBrowse the free reference

See all 19 Integrals formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Riemann Sum?

The definite integral is defined as the limit of Riemann sums. Partition [a,b] into n subintervals of width Δx = (b-a)/n. It is one of the integrals formulas in the CalcRef reference.

What do the symbols in the Riemann Sum mean?

In the Riemann Sum, a is the lower bound; b is the upper bound; n is the number of subintervals.

How do you use the Riemann Sum?

Worked example. Approximate ∫₀¹ x² dx using a right Riemann sum with n = 4. Δx = 1/4. Right endpoints: 1/4, 1/2, 3/4, 1. Sum = (1/16 + 1/4 + 9/16 + 1)(1/4) = (1/16 + 4/16 + 9/16 + 16/16)(1/4). = (30/16)(1/4) = 30/64 = 15/32. Answer: 15/32 = 0.46875 (exact: 1/3 ≈ 0.3333).

Practice

Quiz yourself on Integrals

Riemann Sum is one of 19 Integrals formulas in CalcRef, out of 136 in the library. Today's free round is 10 mixed multiple-choice questions drawn from the same dataset as this page, refreshed every day and replayable as often as you like - no account needed.

Take today's free quizAll 19 Integrals formulas