Double Integral Calculator
Double integral calculator: numerically evaluate ∫∫ f(x,y) dx dy over a rectangular region using Simpson's rule for a fast volume estimate.
Editorial review: Farrow (pseudonym; identity and professional credentials are not disclosed). This page is for educational purposes, not professional advice.
Your inputs
Results update as you type. Nothing you enter leaves your device.
Results
Example with the starting values
These results use the calculator's initial inputs. Change the values above to calculate your own estimate.
Inputs
- f(x,y)
- x·y
- x from
- 0
- x to
- 1
- y from
- 0
- y to
- 1
Results
- Double integral
- 0.25
How does the Double Integral Calculator work?
A double integral ∫∫ f(x,y) dx dy over a rectangle sums the function's value over a grid of x,y points, weighted by Simpson's rule, to estimate the volume under the surface.
Enter the function using x and y (operators +, -, *, /, ^, and sin, cos, sqrt, exp), the x range, and the y range. More grid steps give a more accurate estimate.
This is an iterated integral calculator: it evaluates the inner integral over y first at each x, then sums those slices across x, the same order a double integral solver does by hand.
For example, f(x,y) = x·y over x from 0 to 1 and y from 0 to 1 gives ∫₀¹∫₀¹ x·y dx dy = 0.25, the volume under that surface over the unit square.
Frequently asked questions
What does a double integral calculate?
The volume between a surface z = f(x,y) and the x-y plane over a rectangular region.
How accurate is this calculator?
It uses Simpson's rule on an even grid, which is very accurate for smooth functions with enough steps (the default is 20 per axis).
Can I use any function?
Yes, any expression in x and y built from +, -, *, /, ^ and sin, cos, sqrt, exp, log.
What is Simpson's rule?
A numerical integration method that fits parabolas to the function over small intervals, giving a much closer estimate than simple rectangles.