Double Integral Calculator
Evaluate a double integral of a polynomial in x and y over a rectangular region.
About Double Integral Calculator
Computes the double integral ∬ f(x,y) dA over a rectangle [a,b]×[c,d] using 2D composite Simpson's rule. The method applies Simpson's 1/3 rule independently in both the x and y directions, giving O(h⁴) accuracy in each dimension.
Formula
Double integral
∬[a,b]×[c,d] f(x,y) dA ≈ (hx/3)(hy/3) Σᵢ Σⱼ wᵢ wⱼ f(xᵢ, yⱼ)
- hx
- (b-a)/n
- hy
- (d-c)/n
- wᵢ,wⱼ
- Simpson weights (1,4,2,4,...,1)
Worked example
With the default values below, Double Integral Calculator returns ∬f(x,y) dA of 0.66666667. Change any input above and every figure here is recalculated in your browser.
Inputs used
- f(x, y)
- x^2 + y^2
- x lower (a)
- 0
- x upper (b)
- 1
- y lower (c)
- 0
- y upper (d)
- 1
- Grid points per axisaccepts 4–400
- 50
Results
- ∬f(x,y) dA
- 0.66666667
Step by step
- 1Values used
- 2Double integral
- 3∬f(x,y) dA= 0.66666667
Assumptions and limits
- The result is derived only from the 6 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What regions are supported?
This calculator handles rectangular regions. For non-rectangular regions, you would need to express the bounds as functions of the other variable.
How many grid points do I need?
50 points per axis (2500 evaluations total) is usually sufficient. Increase for rapidly varying functions.