What functions are supported?

You can use any JavaScript-compatible math expression: arithmetic operators (+, -, *, /, **), Math functions (sin, cos, tan, exp, log, sqrt, abs, pow), and constants (Math.PI, Math.E). For example: sin(x)*cos(y) or x**2 + y**2.

How accurate is the result?

Composite Simpson's rule has O(h^4) error per dimension. With the default 100 subdivisions the error is typically below 1e-8 for smooth functions. Increase subdivisions for sharper or more oscillatory integrands.

What does "subdivisions" mean?

Subdivisions (n) is the number of equal sub-intervals along each axis. Simpson's rule requires n to be even. The total number of function evaluations is (n+1)^2, so n=100 means ~10,201 evaluations.

Why does the result show NaN or Infinity?

This usually means the function has a singularity inside the integration region (e.g. 1/x with x-bounds spanning 0). Check that the function is finite everywhere inside [a,b]×[c,d].

Can I integrate over non-rectangular regions?

Not directly — this tool supports only rectangular (box) regions. For general regions, rewrite the integrand with an indicator function, or split the domain into rectangles.

Is my data sent to a server?

No. All computation happens inside your browser using JavaScript. Nothing is transmitted to any server.

Double Integral Calculator

Name: Double Integral Calculator
Availability: InStock
Author: Nham Vu

Numerically compute double integrals over rectangular regions with step-by-step Simpson's rule breakdown — no server required.

Define the Integral

f(x, y) — Integrand

Use x, y, standard operators, and sin, cos, exp, log, sqrt, etc.

x bounds

Lower (a)

Upper (b)

y bounds

Lower (c)

Upper (d)

Subdivisions (n) — must be even

Higher n = more accurate. Range: 2 – 500.

Result

Configure the integral on the left and click Calculate.

Step-by-Step Breakdown

Steps will appear here after calculation.

How Simpson's Rule Works

Composite 2D Simpson's rule approximates ∬∬ f(x,y) dA by dividing each axis into n equal sub-intervals and applying the quadrature weights 1, 4, 2, 4, …, 4, 1 along each dimension.

The formula for a single axis of width h = (b−a)/n is:

(h/3) × [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(xₙ)]

In 2D this is applied first along y (inner integral) for each fixed x node, then along x (outer integral) over the row sums.

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Summary

Numerically compute double integrals over rectangular regions with step-by-step Simpson's rule breakdown — no server required.

How it works

Enter the integrand function f(x, y) using standard math notation (e.g. x*y, sin(x)*cos(y)).
Set the lower and upper bounds for x (a to b) and y (c to d).
Choose the number of subdivisions (higher = more accurate).
Click "Calculate" to run composite 2D Simpson's rule in your browser.
The result appears with a step-by-step breakdown showing inner and outer integral approximations.

Use cases

Verify double integral results from textbook problems.
Quickly estimate the volume under a surface over a rectangular domain.
Check calculus homework answers without a CAS system.
Explore how changing bounds or the function affects the integral value.
Compute probability mass for bivariate distributions over a rectangular region.
Validate numerical integration code by comparing against a known tool.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu