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Multiple Integral Calculator con Pasos

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Looking for related tools? Try our integral calculator with steps, our derivative calculator or our limit calculator with steps.

Sin servidor⚡ Web Worker
Límites de integración (de fuera a dentro)
∫
d
∫
d
f(x,y) =
Cambio de Variables / Jacobian
Teclado matemático
Chargement du clavier…
Ejemplos

Integración común

Reglas básicas

  • Regla de la potencia:
    ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
  • Constante:
    ∫c dx=cx+C\int c \, dx = cx + C
  • Múltiplo constante:
    ∫cf(x) dx=c∫f(x) dx\int c f(x) \, dx = c \int f(x) \, dx

Funciones especiales

  • ∫sin⁡(x) dx=−cos⁡(x)+C\int \sin(x) \, dx = -\cos(x) + C
  • ∫cos⁡(x) dx=sin⁡(x)+C\int \cos(x) \, dx = \sin(x) + C
  • ∫ex dx=ex+C\int e^x \, dx = e^x + C
  • ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C
∫∫

Introduce una expresión y pulsa Go

¿Por qué usar nuestra Calculadora de Integrales Múltiples?

Evalúe de manera intuitiva integrales dobles y triples complejas sobre límites personalizados. Utilice soporte nativo para cambios de coordenadas para mapear variables libremente, aplicando cálculos automáticos de determinantes jacobianos construidos específicamente para dominios polares y esféricos.

Doble (2D)

∬Df(x,y) dA\iint_{D} f(x,y) \,dA

Calculate volume under a 3D surface or the area of 2D regions.

Triple (3D)

∭Ef(x,y,z) dV\iiint_{E} f(x,y,z) \,dV

Calculate total mass, center of mass, and hyper-volumes in 3D space.

Cambio de Variables

dxdydz=∣J∣ dudvdwdxdydz = |J| \,dudvdw

Automatically map geometries via Jacobian determinants (e.g. Polar).

Cómo funciona la integración

La integración múltiple expande el cálculo estándar de una sola variable a volúmenes y espacios n-dimensionales. Esta herramienta evalúa primero las integrales más internas analíticamente paso a paso, sustituye los límites internos en la expresión restante e itera cronológicamente hacia afuera.

Preguntas frecuentes

¿Maneja derivadas segundas y terceras?

¡Sí! Puedes calcular derivadas de orden superior simplemente aplicando el operador de diferenciación varias veces.

¿Son precisos los resultados?

Nuestro motor simbólico utiliza reglas algebraicas formales para garantizar la perfección analítica para todas las funciones compatibles.

How do you evaluate a multiple integral step by step?

A multiple integral is evaluated iteratively from the inside out. Integrate with respect to the innermost variable while treating all other variables as constants. Then substitute the inner boundary limits and repeat the integration for each subsequent outer variable.

What is the difference between a double and a triple integral?

A double integral (2D) integrates a function f(x, y) over a planar region to find surface area, 2D mass, or volume under a surface. A triple integral (3D) integrates f(x, y, z) over a 3D solid region to compute total volume, mass distributions, center of mass, and hyper-quantities.

How does Change of Variables work with the Jacobian determinant?

When transforming coordinates (such as from Cartesian (x, y) to Polar (r, θ)), the differential area element is scaled by the absolute value of the Jacobian determinant |J|. For polar coordinates dx dy = r dr dθ.

Does this calculator support Polar, Cylindrical, and Spherical coordinates?

Yes. Mathsway provides built-in coordinate presets for Polar (2D), Cylindrical (3D), and Spherical (3D) transformations with automatic Jacobian determinant evaluation.

Is this multiple integral calculator free and private?

Yes, it is 100% free with no subscription or account required. All computations run in your web browser via Web Workers, keeping your queries completely private.

How are multiple integrals related to derivatives?

Multiple integrals accumulate quantities over multi-dimensional domains, whereas partial derivatives measure rates of change along individual coordinate axes. You can verify antiderivatives with our integral calculator with steps or calculate derivatives with full steps using our derivative calculator.

How do multiple integrals connect to limits and single-variable calculus?

Multiple integrals are Riemann sums taken as partition sizes approach a limit of zero across multiple dimensions. For exploring boundary conditions or single-variable calculus, use our limit calculator with steps.