Mathematics, Statistics & Geometry

Multiple Integral Calculator

Evaluate complex double integrals (surface area and 3D volume) over rectangular regions using high-precision 2D numerical integration.

Volume (Double Integral)
12
Calculation StepsFunction: f(x,y) = x^2 * y Integrate with respect to x from 0 to 2 Integrate with respect to y from 0 to 3 Applying 2D Numerical Integration (Simpson's 1/3 Rule): Grid Resolution: 40 x 40 evaluated points. Δx = 0.0500, Δy = 0.0750 Result: 12.000000

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Calculating 3D Volumes

The Multiple Integral Calculator brings multivariable calculus to your browser. By employing high-resolution 2D numerical integration grids, it bypasses unsolvable algebraic limits to instantly calculate the exact volume beneath any continuous 3D surface.

V=∬Rf(x,y) dx dy\begin{aligned} V = \iint_{R} f(x,y) \, dx \, dy \end{aligned}

Where:
V=
The total 3D space under the surface
f(x,y)=
The height of the 'roof' at any specific coordinate
R=
The 2D floor area you are integrating over

Slicing the Z-Axis

To understand a double integral, imagine a loaf of bread.

  1. The inner integral (dx) calculates the exact 2D area of a single slice of bread.
  2. The outer integral (dy) takes the area of that slice, multiplies it by the thickness of the slice, and adds up all the slices to get the total volume of the loaf.

Real-World Applications

  • Mechanical Engineering: Calculating the exact Center of Mass (centroid) for irregular, 3D machined parts to ensure they balance perfectly when spinning.
  • Meteorology: Calculating the total volume of rainfall over a specific state by integrating a 3D weather radar topographical map.
  • Electromagnetism: Using Gauss's Law to calculate the total electric flux passing through a 3D Gaussian surface.

Frequently Asked Questions

While a standard single integral finds the 2D 'area under a curve', a double multiple integral finds the 3D 'volume under a surface'.

You solve it from the 'inside out'. You treat y as a constant and integrate with respect to x. Then, you take that result and integrate it with respect to y.

Solving complex double integrals algebraically is often impossible. This calculator uses a 2D variation of Simpson's 1/3 Rule to numerically slice the volume into thousands of tiny rectangular prisms and add them up.

Imagine integrating over a plot of land. The region is the exact boundary of the property (the x and y limits). The function f(x,y) is the height of the hills and valleys on that property.

Yes. If the function inside the double integral is simply '1', the mathematical result is exactly equal to the 2D surface area of the region R.

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