Integrals

An integral accumulates infinitesimal contributions of a quantity to find a total — most commonly, the area under a curve. The definite integral of a function over an interval gives a single number; the indefinite integral gives a family of antiderivative functions.

What is an integral?

Integration is the reverse operation of differentiation: where a derivative breaks a function down into its instantaneous rate of change, an integral builds a total back up from that rate, either as a running accumulation (indefinite) or a specific measured quantity over an interval (definite).

Basic formula

For a function f(x), the definite integral from a to b is written ∫ₐᵇ f(x) dx, and by the Fundamental Theorem of Calculus equals F(b) − F(a), where F is any antiderivative of f.

Example

The integral of x² from 0 to 3 is [x³/3] evaluated from 0 to 3, which equals 27/3 − 0 = 9.

Common mistakes

Forgetting the constant of integration on indefinite integrals; mixing up which bound is upper vs. lower; misapplying the power rule when the exponent is −1 (the integral of 1/x is ln x , not a power of x).
  • Derivatives

Questions about integrals

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