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.

Variables

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)

Practice this and 135 more formulas in the CalcRef workspace — quizzes, reference tables, a 16-category unit converter, and an expression evaluator.