Definition

Definite integral

(F3.2.1) F3.2 · Integration as Accumulation →

\int_{a}^{b} f (x) d x = N \to \infty lim i = 1 \sum N f (x_{i}) Δ x

Statement

The limiting value of a sum of thin rectangular strips under the curve of f, between a and b.

Symbols and units

$f (x)$	the function being integrated	—
$a, b$	limits of integration	—
$Δ x$	strip width	—

When it applies

Functions that are continuous (or piecewise continuous) on [a, b].

Don't use when

Requires the integrand to be (piecewise) continuous on the interval; an unbounded discontinuity inside [a, b] needs an improper integral treatment instead.

Derivation

Divide [a, b] into N thin strips of width Δx and height f(x_i):

i = 1 \sum N f (x_{i}) Δ x

Let N → ∞ (Δx → 0); the sum converges to the exact area:

\int_{a}^{b} f (x) d x = N \to \infty lim i = 1 \sum N f (x_{i}) Δ x

Worked examples

Displacement from a changing velocity

v(t) = 2t (m/s), from t = 0 to t = 3 s:

Δ x = \int_{0}^{3} 2 t d t = [t^{2}]_{0}^{3} = 9 m

Unlike constant velocity, the accumulated distance grows with the square of time — exactly what the integral captures.

Watch out for

Integration and differentiation are inverse operations, but only where the integrand is continuous over the interval.

Connects to

Derivative