Double Integral Calculator
Evaluate a double integral of a polynomial in x and y over a rectangular region.
Inputs
Enter function using x and y.
∬f(x,y) dA
0.66666667
Step by step
Values 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 axis = 50
Double integral
∬[a,b]×[c,d] f(x,y) dA ≈ (hx/3)(hy/3) Σᵢ Σⱼ wᵢ wⱼ f(xᵢ, yⱼ)
∬f(x,y) dA
= 0.66666667
How it works
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)
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.