Double Integral Calculator: Iterated Integrals Step by Step

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Formula at a glance

Double Integral Calculator: Iterated integral (Fubini) — ∬_R f dA = ∫ₐᵇ ( ∫_{g₁(x)}^{g₂(x)} f(x, y) dy ) dx. Example: ∫₀¹∫₀ˣ xy dy dx → Double integral: 1/8 ≈ 0.125 — How the Double Integral Calculator works: Iterated integral (Fubini) — ∬_R f dA = ∫ₐᵇ ( ∫_{g₁(x)}^{g₂(x)} f(x, y) dy ) dx

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📘 About the Double Integral Calculator

A double integral adds up a function of two variables over a region of the plane — the volume under a surface z = f(x, y). With f = 1 it gives the area of the region; in physics it gives mass, centre of mass and moments of inertia.

Enter the function and the limits. The inner limits may depend on the outer variable, so you can integrate over triangles, parabolic regions and parts of circles. The calculator integrates the inner variable first (treating the other as a constant), substitutes the limits, then integrates again, and confirms the result with Gauss–Legendre quadrature.

🧮 Formula

Iterated integral (Fubini)

∬_R f dA = ∫ₐᵇ ( ∫_{g₁(x)}^{g₂(x)} f(x, y) dy ) dx

Area of a region

A = ∬_R 1 dA

✏️ Worked examples

∫₀¹∫₀ˣ xy dy dx

You enter
Function f(x, y)	x*y
Order of integration	dy dx (y inside, x outside)
Inner lower limit	0
Inner upper limit	x
Outer lower limit	0
Outer upper limit	1

You get
Double integral	1/8 ≈ 0.125
Decimal value	0.125
Region	0 ≤ y ≤ x, 0 ≤ x ≤ 1

1/8 ≈ 0.125.

∬ (x² + y²) over the unit square

You enter
Function f(x, y)	x^2+y^2
Order of integration	dy dx (y inside, x outside)
Inner lower limit	0
Inner upper limit	1
Outer lower limit	0
Outer upper limit	1

You get
Double integral	2/3 ≈ 0.6666666667
Decimal value	0.666666666667
Region	0 ≤ y ≤ 1, 0 ≤ x ≤ 1

2/3 ≈ 0.6666666667.

🧭 How to use this calculator

What you enter

Function f(x, y)
Order of integration
Inner lower limit
Inner upper limit
Outer lower limit
Outer upper limit

What you get

Double integral
Decimal value
Region

❓ Frequently asked questions

How do you evaluate a double integral?

Integrate the inner variable first with the other held constant, substitute its limits, then integrate the result with respect to the outer variable.

Does the order of integration matter?

For continuous functions the answer is the same (Fubini’s theorem), but the limits change when you switch order and one order is often easier.

How do I find the area of a region with a double integral?

Integrate f = 1 over the region. For the region between y = x² and y = x from x = 0 to 1 the area is 1/6.

∬ Double Integral Calculator (Iterated Integrals)