Double Integral Calculator

How the Double Integral Calculator Works

Enter your function of x and y, set limits for both variables and choose order of integration. The calculator works the inner integral first, treating other variable as a constant, then integrates result with respect to outer variable, displaying both stages in full.

The notation ∬ f(x, y) dA means integrating a function over a region of plane rather than an interval of a line. In practice every double integral becomes an iterated integral, two ordinary integrals performed one after other, which is exactly how calculator and you in an exam, actually evaluate them.

Worked Example

Evaluate ∬ xy dA over the rectangle where x runs from 0 to 2 and y runs from 1 to 3:

• Inner integral with respect to y, holding x constant: ∫ xy dy from 1 to 3 = x[y²/2] from 1 to 3 = x(9/2 - 1/2) = 4x
• Outer integral: ∫ 4x dx from 0 to 2 = [2x²] from 0 to 2 = 8

The volume under surface z = xy over that rectangle is 8. The pattern generalises, inner integral first, its result becomes integrand of the outer one.

Choosing the Order of Integration

Over a rectangle with constant limits, Fubini's theorem says order does not change answer, so integrate in whichever order is easier. Over a general region limits of inner integral depend on outer variable and here order becomes a genuine strategic choice, a region described as y running from x² to x needs different limits than same region sliced other way.

Two situations force decision. Some regions split into multiple pieces in one order but stay whole in other, doubling or halving work. And some integrands can only be integrated in one order at all, the classic e^(y²) has no elementary antiderivative in y, so any integral containing it must run the x integration first. When an integral looks impossible, reversing the order is standard rescue move and sketching region is step that makes new limits readable.

Polar Coordinates and the Extra r

Circular regions fight against x and y limits but surrender instantly to polar coordinates, where x = r cos θ and y = r sin θ. The conversion carries one rule that examiners love and students forget, the area element dA becomes r dr dθ, not just dr dθ. That extra r is Jacobian of transformation and omitting it is single most common error in polar double integrals.

The payoff is real, integrating over a disc of radius 2 becomes r from 0 to 2 and θ from 0 to 2π with no square roots in sight and expressions like x² + y² collapse to r². Any region with circles, sectors or annuli in its description should trigger the polar conversion reflex.

What Double Integrals Are For

Beyond volume, applications follow one template, something varies across a region and double integral totals it. Integrating a density function over a plate gives its mass. Integrating x times density, divided by mass, locates centre of mass. Integrating function and dividing by region's area gives average value, whether that is average temperature over a surface or average rainfall over a catchment. And integrating 1 over a region returns region's area itself, which is occasionally cleanest way to find an awkward area. These are the standard second year university and further maths applications and they all reduce to the same iterated procedure calculator demonstrates.