What is a double integral?
An integral of a function of two variables over a 2D region.
Double Integral Calculator
Integrate a constant density or the product xy over an axis-aligned rectangle.
Enter the region and integrand
Choose integrand type, then provide rectangle bounds.
About this calculator
How to use Double Integral Calculator, plus the formulas and logic behind the results.
Tool description
Over a rectangle, iterated integrals separate: ∬ f(x,y) dA = ∫_a^b ∫_c^d f(x,y) dy dx.
How to use
- Select constant or xy integrand.
- Enter the rectangle bounds [a,b]×[c,d].
- Select Calculate to view the double integral value.
- Interpret constant mode as k times the area of the rectangle.
Formulas & reference
∬_R k dA = k (b−a)(d−c)∬_R xy dA = [(b²−a²)/2] [(d²−c²)/2]
Double integral tips
- Order of integration can be swapped on rectangles
- Constant integrand → volume of a box
- xy separates into a product of 1D integrals
- Check bound order if the result looks negative
Calculation logic
Evaluates the separated iterated integral in closed form for the two supported integrands.
Rectangular regions only—no general curved domains.
Double Integral Calculator FAQ
Common questions about this calculator and how to use the results.
Why rectangles only?
They allow simple exact formulas for these educational cases.
What does ∬ k dA mean?
k times the area of the region.
What about ∬ xy?
It factors into (∫ x dx)(∫ y dy) on a rectangle.
Can bounds be reversed?
Yes—signs follow the oriented intervals.
Is this free?
Yes.