Double Integral Calculator: Free Step-by-Step Tool 2026

Q: How do I find the limits of a double integral?

Integrate in y first: its limits can depend on x, the curves bounding the region. The outer x limits are the constants where the region starts and ends.

Q: Can the y-limits of a double integral be functions of x?

Yes. Enter expressions such as 0 to x or 0 to 1-x to integrate over triangles and other non-rectangular regions.

This free double integral calculator evaluates an iterated integral ∬∬ f(x, y) dA over a region you choose. Enter the integrand and the limits — the inner (y) limits may depend on x — and it returns the value plus a plot of the region, so it works for rectangles, triangles and other shapes, not just boxes.

∬∬ f(x, y) =

x fromto

y fromto

y limits may use x — e.g. 0 to x for a triangle

Double integral

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Enter the integrand and limits, then press Calculate.

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What the double integral calculator computes

A double integral $\iint_R f(x,y)\,dA$ adds up the values of a function over a flat region $R$ in the xy-plane. When $f=1$ it gives the area of the region; with other integrands it gives the signed volume under the surface $z=f(x,y)$, or totals like mass and average value. This calculator evaluates it as an iterated integral — integrating in $y$ first, then $x$ — using fast numerical integration, and draws the region so you can check your limits at a glance.

Setting up the limits

Work from the inside out. The inner ($y$) limits may depend on $x$; the outer ($x$) limits are constants:

Region	x from / to	y from / to
Rectangle	0 to 1	0 to 1 (constants)
Triangle (under y = x)	0 to 1	0 to x
Triangle (under x + y = 1)	0 to 1	0 to 1−x

Worked example: area of a triangle

The triangle with vertices $(0,0)$, $(1,0)$, $(1,1)$ lies under the line $y=x$, so $$\int_{0}^{1}\!\int_{0}^{x} 1 \; dy\,dx = \int_0^1 x\,dx = \tfrac{1}{2}.$$ Tap "area of a triangle" above and the calculator returns $0.5$ with the triangle shaded.

Fubini's theorem and order of integration

Fubini's theorem says that for a well-behaved function you can integrate in either order — $dy\,dx$ or $dx\,dy$ — and get the same answer. You pick the order that makes the limits simplest. This tool integrates $dy$ first; to swap the order, redescribe the region with $x$ as the inner variable and relabel.

⚠ Polar coordinates. For circles and rings, converting to polar coordinates $(r,\theta)$ usually makes the limits far simpler — but the area element becomes $dA = r\,dr\,d\theta$. Include that extra factor of $r$ in your integrand when you set it up here.

What double integrals are used for

Area of a 2-D region (integrate $f=1$).
Volume under a surface $z=f(x,y)$.
Mass, center of mass and moments for a flat plate of given density.
Probability — integrating a joint density over a region gives a probability, which is everywhere in statistics and machine learning.

For the next step up, see our triple integral calculator; for the single-variable basics, the integral calculator. For the formal theory, the multiple integral article on Wikipedia is a good reference.

Frequently asked questions

What is a double integral? It is an integral of a function of two variables over a flat region, written $\iint_R f(x,y)\,dA$. With $f=1$ it gives the region's area; otherwise it gives the volume under the surface.

How do I find the limits of a double integral? Integrate in $y$ first: its limits can depend on $x$ (the curves bounding the region top and bottom). The outer $x$ limits are the constants where the region starts and ends.

Can the y-limits be functions of x? Yes. Enter expressions such as 0 to x or 0 to 1-x to integrate over triangles and other non-rectangular regions.

How accurate is the calculator? It uses nested Simpson's-rule integration, which is accurate to several decimal places for smooth integrands.