Double integral calculator
This double integral calculator evaluates ∬ f(x, y) dA numerically: over a rectangle, over a region whose inner limits depend on the outer variable (such as 0 ≤ y ≤ x) or in polar coordinates with r dr dθ. Example: x² + y² over 0 ≤ x ≤ 2, 0 ≤ y ≤ x gives 16/3 = 5.333333.
Source: Wikipedia, Multiple integral: Fubini’s theorem, normal domains, polar coordinates dA = r dr dθ (read 15.10.2026). Updated: .
How it is calculated
How a double integral is calculated
A double integral ∬ f(x, y) dA adds up f over a region of the plane – the volume under the surface z = f(x, y). By Fubini’s theorem it is computed as two single integrals, one inside the other. For the region 0 ≤ x ≤ 2, 0 ≤ y ≤ x the inner integral runs over y with x held fixed: ∫₀ˣ (x² + y²) dy = x³ + x³/3 = 4x³/3. The outer integral then gives ∫₀² 4x³/3 dx = 16/3 = 5.333333.
Examples
| f | Outer | Inner | Result |
|---|---|---|---|
| x² + y² | 0 ≤ x ≤ 2 | 0 ≤ y ≤ x | 5.333333 (= 16/3) |
| xy | 0 ≤ x ≤ 1 | x² ≤ y ≤ x | 0.04166667 (= 1/24) |
| sin(x)cos(y) | 0 ≤ x ≤ π/2 | 0 ≤ y ≤ π/2 | 1 |
| exp(x²) | 0 ≤ y ≤ 1 | y ≤ x ≤ 1 | 0.8591409 (= (e − 1)/2) |
| x² + y² | 0 ≤ θ ≤ 2π | 0 ≤ r ≤ 1 | 1.570796 (= π/2) |
Changing the order of integration
The fourth example shows why the order matters. e^(x²) has no elementary antiderivative, so dx first is hopeless by hand. The same triangle described the other way, 0 ≤ x ≤ 1 and 0 ≤ y ≤ x, gives ∫₀¹ x·e^(x²) dx = (e − 1)/2 = 0.8591409. The calculator works numerically and handles both orders; switch between dy dx and dx dy to check your own bounds.
Polar coordinates
For discs, rings and sectors use polar mode: x = r cos θ, y = r sin θ and dA = r dr dθ. The extra factor r is the Jacobian and is added automatically – enter only f. The area of a disc of radius 2 is ∬ 1 dA = 4π = 12.56637. For integrals in one variable use the integral calculator, for slopes the derivative calculator, and to look at a function first the graphing calculator.
Frequently asked questions
How do you calculate a double integral?
Integrate the inner variable first while the outer one is held constant, then integrate the result over the outer variable (Fubini’s theorem).
Does the order dy dx or dx dy matter?
Not for the value, as long as the region is described correctly. It can make the calculation by hand much easier, as with e^(x²) on a triangle.
How do I enter a region that is not a rectangle?
Make the inner limits depend on the outer variable, for example 0 ≤ x ≤ 1 and x² ≤ y ≤ x for the region between the parabola and the line.
Why is there an extra r in polar coordinates?
A small polar cell has the area r · dr · dθ. The calculator multiplies by r for you; for the area of a unit disc enter f = 1 and get π.
How accurate is the result?
The calculator compares two runs with 16 and 32 subintervals of the 10-point Gauss–Legendre rule. For smooth functions the estimated error is usually below 10⁻¹².
Sources and legal basis
- Wikipedia, Multiple integral: Fubini’s theorem, normal domains, polar coordinates dA = r dr dθ (read 15.10.2026)
- NIST DLMF §3.5(v), Table 3.5.2: nodes and weights of the 10-point Gauss–Legendre formula (read 15.10.2026)
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