maths.free › Calculus › Definite integrals

Definite integrals

Area under a curve and the fundamental theorem of calculus.

A definite integral is the signed area between the curve and the axis on an interval. Find an antiderivative F, then compute F(b) − F(a) — the fundamental theorem of calculus. The shaded region in the graph is the area being computed.

Жұмыстағы мысал: integrate x^2 dx from 0 to 1

Calculus · Definite-Integrals

Integrate x^2 from 0 to 1

\int_{0}^{1} x^{2}\, dx

Қадам сайын

  1. \int_{0}^{1} x^{2}\, dx

    First find an antiderivative F, then evaluate F(b) − F(a).

  2. \int x^{2}\, dx = \frac{x^{3}}{3}

    Power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1) (n ≠ −1).

  3. F(1) - F(0) = \left(\frac{1}{3}\right) - \left(0\right)

    Fundamental theorem of calculus: plug in the limits.

  4. = \frac{1}{3}

    Simplify.

Жауап беріңіз
\frac{1}{3}

Өзіңіздіңіңізді сынап көріңіз

Келесіде Calculus

LimitsDerivativesIntegralsTaylor seriesSeries and sumsMaxima and minima