Integrals

The integral as accumulated area — Riemann sums, definite integrals, and the antiderivative as an operation that reverses differentiation.

Practice exercises →

Riemann sum approximation of ∫₀³ x² dx = 9 — drag slider to add more rectangles

n = 480

Error: 3.0938 — approaches 0 as n → ∞

Definition

The definite integral of $f$ from $a$ to $b$ is the signed area between the graph of $f$ and the $x$ -axis over the interval $[a, b]$ :

$\int_a^b f(x)\,dx$

Area above the axis counts positive; area below counts negative.

To compute it, we approximate using Riemann sums — the sum of $n$ thin rectangles of width $\Delta x = (b-a)/n$ and heights $f(x_i^*)$ :

$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\,\Delta x$

Computing from the definition

$\int_0^2 3\,dx$ is the area of a rectangle with width $2$ and height $3$ : the answer is $6$ .

$\int_{-1}^1 x\,dx = 0$ because the positive area above the axis (for $x > 0$ ) exactly cancels the negative area below (for $x < 0$ ) — the integrand is odd.

Compute the integral

Evaluate $\displaystyle\int_0^\pi \sin x\,dx$ .

Related concepts

Needs first

Derivatives Limits

Applications of Integration Fundamental Theorem of Calculus Integration by Parts Lebesgue Integral Riemann Sum Techniques of Integration

Uses this

Fourier Transform

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