Calculadora de integrais

How to calculate the integral

1. Insira a função f(x). Por exemplo x^2 or sin(x).
2. Insira os limites. Os limites inferior e superior definem o intervalo sobre o qual a área é calculada.
3. Calcular. You get the value of the definite integral, i.e. the area between the curve and the x-axis.

What does the definite integral mean?

The definite integral ∫ₐᵇ f(x) dx describes the signed area between the function's curve and the x-axis over the interval [a, b]. When the curve is above the x-axis, the area is positive; below the axis the area is counted as negative. The integral is a central tool in calculating areas, volumes, work and many physical quantities.

Simpson's rule

A calculadora usa a regra de Simpson, que estima a área ajustando arcos parabólicos à curva. O método é muito preciso para funções suaves e fornece um resultado praticamente correto para problemas comuns. Se a função tiver uma descontinuidade ou uma assíntota vertical no intervalo escolhido, o resultado pode ser pouco confiável — nesse caso, divida o intervalo em partes.

Exemplo: distância a partir da velocidade

If velocity as a function of time is v(t) = 2t (m/s), the distance travelled over t = 0…5 is the definite integral of velocity ∫₀⁵ 2t dt. Geometrically this is the area of the triangle under the velocity graph: base 5 and height v(5) = 10, so the area = ½ · 5 · 10 = 25 metres. The calculator gives the same resultado numerically. Thus the integral connects velocity and distance in the same way that the derivative connects distance and velocity — they are inverse operations of each other.

Definite and indefinite integral

The indefinite integral ∫ f(x) dx means a function whose derivative is f(x); it always involves a constant term + C. The definite integral ∫ₐᵇ f(x) dx is a number obtained by substituting the limits and subtracting. The fundamental theorem of calculus ties these together: the definite integral is the difference of the antiderivative's values at the limits. This calculator focuses on the definite integral, i.e. the area over a given interval, which is enough for most practical and school problems.

De área a volume e média

The definite integral calculates the area under the curve, but much else follows from the same idea. When the curve is rotated around an axis, the integral gives the volume of the resulting solid — this is how the volumes of solids of revolution are calculated, for example. In physics the integral turns velocity into distance and force into work done, because these are "accumulations" over time or distance.

The integral is also used to calculate the average of a function over an interval: it is the integral divided by the length of the interval. This gives, for example, the average of a varying temperature or power over a certain period. In all of these the same idea of continuous summation recurs: the integral adds up infinitely many small parts. This calculator gives the area numerically, and once you recognise what the area represents in each situation, the same tool serves many kinds of problems.

Good to know about the integral

• A definite integral is a number, an indefinite integral is a function (+ constant C).
• If the upper and lower limits are the same, the value of the integral is zero.
• Swapping the limits changes the sign of the integral.
• A parte abaixo do eixo x é contada como área negativa.

Em resumo

The integral calculator computes the definite integral, i.e. the area under the curve, with the given limits. Enter the function and the limits, and the tool computes the resultado accurately with Simpson's rule. The same idea of area generalises to volumes, work and averages, so the calculator serves many kinds of problems.

Perguntas frequentes