CalcOpenly
Português

Calculadora de integrais definidas

The definite integral of any function f(x) between two limits, by adaptive Gauss–Kronrod quadrature to about 20 significant digits, with the area shaded.

Atualizado Exemplos verificados: 7

Use x as the variable, e.g. x^2 * sin(x), 1/x, sqrt(x). Trig functions use radians here.
Numbers or expressions like pi/2
Experimentar
Integrais
Integrais: 0.746824132812
Máximo de casas decimais: 12; Ao mais próximo; empates afastando-se de zero
Estimated error
1.1 × 10⁻²¹
Function evaluations
45
Subintervals used
2

The signed area between f(x) and the x-axis from 0 to 1 is 0.7468241328; area below the axis counts as negative. It took 45 evaluations over 2 subintervals.

Area under f(x) from 0 to 1

00.5100.250.50.751xf(x)ab
Subintervals Linhas: 2
DeParaContributionError estimate
00.50.46133.2 × 10⁻²²
0.510.28558 × 10⁻²²
Como é calculado S

  1. Scale each piece to [−1, 1]

    ∫αβf(x) dx=β−α2∫−11f ⁣(α+β2+β−α2 t)dt\int_{\alpha}^{\beta} f(x)\,dx = \frac{\beta-\alpha}{2}\int_{-1}^{1} f\!\left(\tfrac{\alpha+\beta}{2} + \tfrac{\beta-\alpha}{2}\,t\right)dt

  2. Kronrod 15-point rule with its embedded Gauss 7-point rule

    K15=β−α2∑i=115wif(xi),G7=β−α2∑j=17vjf(x2j),ε=ρ min⁡ ⁣(1,(200 ∣K15−G7∣ρ)3/2)K_{15} = \frac{\beta-\alpha}{2}\sum_{i=1}^{15} w_i f(x_i),\qquad G_7 = \frac{\beta-\alpha}{2}\sum_{j=1}^{7} v_j f(x_{2j}),\qquad \varepsilon = \rho\,\min\!\left(1, \left(\frac{200\,|K_{15} - G_7|}{\rho}\right)^{3/2}\right)

    The 7 Gauss nodes are reused by the 15-point rule, so the error estimate costs no extra evaluations; ρ is ∫|f − mean| over the piece (QUADPACK's scaling). Nodes and weights are carried to 52 digits.

  3. Adaptive bisection

    ∑ε=1.13×10−21≤max⁡(10−40, 10−20 ∣I∣)\sum \varepsilon = 1.13 \times 10^{-21} \le \max(10^{-40},\ 10^{-20}\,|I|)

    Split the subinterval with the largest error estimate until the total met the tolerance: 2 subintervals, 45 evaluations of f.

  4. Resultado

    ∫01f(x) dx≈0.7468241328124270253994674\int_{0}^{1} f(x)\,dx \approx 0.7468241328124270253994674

Sobre Calculadora de integrais definidas

A definite integral ∫ₐᵇ f(x) dx is the signed area between f and the x-axis from a to b, with area below the axis counted as negative. The calculator uses the 15-point Gauss–Kronrod rule from QUADPACK (routine QK15), which samples f at 15 points on each piece and compares the result with the embedded 7-point Gauss rule to estimate its own error. The piece with the largest error is split in two until the total estimated error falls below 10⁻²⁰ of the answer.

It suits integrals with no convenient antiderivative, such as the bell curve e^(−x²), and checking answers found by hand. The default, ∫₀¹ e^(−x²) dx, equals (√π/2)·erf(1) ≈ 0.7468241328 and took 45 evaluations of f over 2 subintervals.

Trig functions use radians, and limits accept expressions such as pi/2. Integrable endpoint singularities, like √x or ln x at 0, are handled because Gauss–Kronrod nodes never touch the endpoints. Swapping the limits flips the sign.

Exemplos resolvidos

∫₀¹ e^(−x²) dx

Function f(x)
exp(-x^2)
Lower limit a
0
Upper limit b
1
Integrais
0.746824132812

Fonte de verificação: (√π/2)·erf(1): Python decimal, erf by its Maclaurin series at 70 digits (hp.py)

∫₀^π sin x dx

Function f(x)
sin(x)
Lower limit a
0
Upper limit b
pi
Integrais
2

Fonte de verificação: −cos π + cos 0 = 2 (antiderivative)

∫₁² 1/x dx

Function f(x)
1/x
Lower limit a
1
Upper limit b
2
Integrais
0.69314718056

Fonte de verificação: ln 2: Python Decimal(2).ln()

∫₀³ x⁵eˣ dx

Function f(x)
x^5 * exp(x)
Lower limit a
0
Upper limit b
3
Integrais
1,686.671880008638

Fonte de verificação: Antiderivative eˣ(x⁵ − 5x⁴ + 20x³ − 60x² + 120x − 120) evaluated in Python decimal

Perguntas

What does a definite integral represent?

The signed area between the curve and the x-axis over an interval: area above the axis counts as positive and area below as negative. ∫₀^π sin x dx = 2 because one arch of the sine curve encloses an area of 2, while ∫₀^2π sin x dx = 0 because the second arch lies below the axis and cancels the first. It is also a total change: integrating speed over time gives distance traveled.

How do you evaluate a definite integral by hand?

Find an antiderivative F with F′ = f, then subtract: ∫ₐᵇ f(x) dx = F(b) − F(a), which is the fundamental theorem of calculus. For ∫₁² 1/x dx, F(x) = ln x, so the answer is ln 2 − ln 1 ≈ 0.693147. When no elementary antiderivative exists, as for e^(−x²), a numerical rule like this one is the practical route.

What is Gauss–Kronrod quadrature?

A rule that estimates an integral from a weighted sum of f at carefully placed points. The 7-point Gauss rule integrates every polynomial up to degree 13 exactly; Kronrod's extension adds 8 more points and reuses the first 7, so comparing the two estimates gives an error bound without extra evaluations. QUADPACK's QK15 routine, used here, pairs it with repeated splitting of the worst subinterval.

What is the integral of e^(−x²)?

It has no antiderivative among elementary functions, so it is written with the error function: ∫₀ˣ e^(−t²) dt = (√π/2) erf(x). From 0 to 1 the value is about 0.7468241328, and over the whole real line it is √π ≈ 1.7724539. This Gaussian integral is what makes the normal distribution's total probability equal 1.

Can a definite integral be negative?

Yes, when more of the area lies below the x-axis than above it, or when the limits run backward. ∫₁² (−x) dx = −1.5, and the integral of x² from 1 down to 0 is −1/3, the negative of ∫₀¹ x² dx = 1/3. For the total area regardless of sign, integrate the absolute value instead: ∫₀^2π |sin x| dx = 4, entered as abs(sin(x)).

Qual é a precisão de “Calculadora de integrais definidas”?

A precisão depende dos dados inseridos e das hipóteses do método. O cálculo decimal usa 50 algarismos significativos, mas estimativas, métodos numéricos e dados de origem podem ter menor precisão; o arredondamento exibido não elimina essas limitações. Exemplos resolvidos verificados com fontes independentes: 7. Por exemplo, “∫₀¹ e^(−x²) dx” é verificado com (√π/2)·erf(1): Python decimal, erf by its Maclaurin series at 70 digits (hp.py).

De onde vem o método?

Piessens, de Doncker-Kapenga, Überhuber, Kahaner — QUADPACK (1983), routine QK15; Wolfram MathWorld — Gauss-Kronrod Quadrature; Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997).

Sobre esta calculadora

∫abf(x) dx≈b−a2∑i=115wi f(ti)ti=a+b2+b−a2 xi\begin{aligned} \int_a^b f(x)\,dx &\approx \frac{b-a}{2}\sum_{i=1}^{15} w_i\, f(t_i) \\[4pt] t_i &= \frac{a+b}{2} + \frac{b-a}{2}\,x_i \end{aligned}

Fontes

  1. Piessens, de Doncker-Kapenga, Überhuber, Kahaner — QUADPACK (1983), routine QK15
  2. Wolfram MathWorld — Gauss-Kronrod Quadrature
  3. Laurie, Calculation of Gauss-Kronrod quadrature rules, Math. Comp. 66 (1997)

Verificado com as referências

Esta calculadora inclui 7 exemplos resolvidos com respostas de fontes independentes. Eles fazem parte do conjunto de testes e você também pode executá-los aqui.

Calculadoras relacionadas