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Multiple Integral Calculator with Steps

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

No backend⚡ Web Worker
Integration Bounds (Outer to Inner)
∫
d
∫
d
f(x,y) =
Change of Variables / Jacobian
Math Keyboard
Chargement du clavier…
Examples

Common Integration

Basic Rules

  • The Power Rule:
    ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
  • Constant:
    ∫c dx=cx+C\int c \, dx = cx + C
  • Constant Multiple:
    ∫cf(x) dx=c∫f(x) dx\int c f(x) \, dx = c \int f(x) \, dx

Special Functions

  • ∫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
∫∫

Enter expression and press Go

Why use our Multiple Integral Calculator?

Evaluate complex double and triple integrals over customized bounds intuitively. Utilize native support for coordinate changes to map variables freely, applying automatic Jacobian determinant calculations specifically built for Polar and Spherical domains.

Double (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.

Change of Variables

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

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

How Integration Works

Multiple integration expands standard single-variable calculus to volumes and n-dimensional spaces. This tool evaluates innermost integrals analytically step-by-step first, substitutes the inner bounds into the remaining expression, and iterates chronologically outward. If a Change of Variables is required (such as shifting (x,y) to Polar (r,θ)), it will automatically differentiate your mapping matrix to derive and multiply the proper Jacobian scale factor before initiating iterations.

Frequently Asked Questions

Does this handle second and third derivatives?

Yes! You can calculate higher-order derivatives by simply applying the differentiation operator multiple times.

Are the results accurate?

Our symbolic engine uses formal algebraic rules to guarantee analytical perfection for all supported functions.

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.