What is a double integral?
A double integral extends the concept of a single integral to functions of two variables. Written as ∬f(x,y) dx dy, it sums the values of a function over a two-dimensional region, computing quantities like volume under a surface, total mass, or area-weighted averages. See also our Derivative Calculator.
How do you evaluate a double integral?
A double integral over a rectangular region is evaluated as an iterated integral: first integrate f(x,y) with respect to y (treating x as constant) from c to d, then integrate the result with respect to x from a to b. Fubini's theorem guarantees this works for continuous functions on rectangles.
What does a double integral represent geometrically?
When f(x,y) ≥ 0, the double integral represents the volume of the solid between the surface z = f(x,y) and the xy-plane over the integration region. For functions that take negative values, it gives the signed volume (positive volume minus negative volume).
What if the integration region isn't a rectangle?
For non-rectangular regions, the bounds of the inner integral become functions of the outer variable. For example, ∫∫ over a triangle might have y bounds from 0 to x. These are called Type I or Type II regions. This calculator handles rectangular regions; for general regions the bounds must be set up carefully. You might also find our Volume of Revolution Calculator useful.
Can the order of integration be switched?
Yes. For continuous functions on a rectangle, Fubini's theorem states that ∫∫f(x,y) dx dy = ∫∫f(x,y) dy dx — the order of integration can be swapped without changing the result. For non-rectangular regions, switching order requires rewriting the limits.
What is the average value of a function over a region?
The average value of f(x,y) over a region R with area A is given by (1/A)·∬f(x,y) dA. This calculator displays the average value alongside the integral result so you can interpret the output in context.
How accurate is the numerical integration?
This calculator uses the midpoint rule (Riemann sum) over a grid of n×n points. With 100 steps the error is typically very small for smooth functions. For higher accuracy, select 200 or 400 steps. Functions with sharp spikes or discontinuities near the bounds may require higher precision settings.
What are common mistakes when setting up double integrals?
Common errors include swapping the lower and upper bounds (making a > b), forgetting that Jacobian factors appear when changing variables, and misidentifying the region of integration for non-rectangular domains. Always verify that your bounds correctly describe the region before computing. Check out our calculate Limit as well.