Double Integral Calculator 2026
Evaluate 2D definite integrals numerically. Instantly compute the volume under a surface using the Trapezoidal rule, configure integration bounds, and adjust subdivision precision for multivariable functions.
Integration Configuration
Define your function, bounds, and numerical precision
Select a predefined function or enter a custom mathematical expression
Number of grid intervals along the X-axis
Number of grid intervals along the Y-axis
Integration Profile
Numerical Approximation & Grid Metrics
Configure your function and bounds, then click Evaluate Integral to compute the volume under the surface.
Integration Algorithms
Standard numerical methods for approximating double integrals over rectangular regions.
| Method | Error Order | Description |
|---|---|---|
| 2D Trapezoidal | O(h²) | Approximates surface using planar trapezoids across the grid. |
| 2D Midpoint | O(h²) | Evaluates function at the center of each grid cell for stability. |
| 2D Simpson’s | O(h⁴) | Uses quadratic polynomials for higher accuracy (requires even subdivisions). |
| Monte Carlo | O(1/√N) | Uses random sampling; effective for high-dimensional or complex regions. |
Multivariable Calculus FAQ
Learn more about double integrals, numerical methods, and evaluating functions of two variables.
A double integral is an integral of a function of two variables over a region in the xy-plane. Geometrically, it represents the volume under the surface defined by the function f(x, y) and above the region of integration.
The 2D Trapezoidal rule approximates the volume under a surface by dividing the region into a grid of rectangles. It evaluates the function at the corners of each rectangle and applies weighted averages, effectively approximating the surface with planar trapezoids.
Fubini’s Theorem states that a double integral over a rectangular region can be evaluated as an iterated integral. This means you can integrate with respect to x first, then y (or vice versa), yielding the same result, provided the function is continuous.
To improve accuracy, increase the number of subdivisions (n and m) for the X and Y axes. A finer grid reduces the step size (dx and dy), which decreases the truncation error of the Trapezoidal or Midpoint approximation.
This calculator is optimized for rectangular regions defined by constant bounds. For non-rectangular regions where the bounds of y depend on x (e.g., c(x) to d(x)), you would need to use an iterated integral approach with variable limits, which requires symbolic computation.