The formula this page uses
∬_R f(x, y) dA = ∫ₐᵇ ∫_{g₁(x)}^{g₂(x)} f(x, y) dy dx
← Back to calculators
Double Integral Calculator
Solve double integral problems with clear steps, notation, and a final check.
∫
Calculate without using AI.
Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
Choose an example or enter a formula.
Show your work Step-by-step method
- No account, AI request, or paid token is used.
Recent calculations
No calculations yet.
Calculus
Double Integral Calculator explained
The short version
- A double integral adds up a quantity over a flat region instead of along a line.
- You do two ordinary integrals, one after the other, and the inner one always finishes first.
- If the region is not a rectangle, the inner limits contain the outer variable.
What each part means
SymbolWhat it means
f(x, y) — The integrandWhat is being added up: a height, a density, or 1 if you only want the area.
R — The regionThe flat patch you are integrating over, described by the four limits.
inner limits — Limits on the first integralThey may contain the outer variable. They must never contain their own variable.
outer limits — Limits on the second integralAlways plain constants. If they are not, the two limits were written in the wrong order.
Show your work: a full example
- A rectangular region∫₀² ∫₁³ 6x²y dy dx, so x runs 0 to 2 and y runs 1 to 3
- Do the inner integral in y, holding x still6x² · y²/2 evaluated from y = 1 to y = 3
- Substitute the y limits3x²(3² − 1²) = 3x²(9 − 1) = 24x²
- Now the outer integral in x∫₀² 24x² dx = 8x³ evaluated from 0 to 2
- Substitute the x limits8(2)³ − 8(0)³ = 64 − 0
- The answer64
A second, different case
- A different case: a triangle, not a rectanglethe region under y = x from x = 0 to x = 2, with f(x, y) = x + y
- Write the limits∫₀² ∫₀ˣ (x + y) dy dx, and notice the inner top limit is x
- Integrate in y, holding x stillxy + y²/2 evaluated from y = 0 to y = x
- Substitutex(x) + x²/2 − 0 = x² + x²/2 = 3x²/2
- Now integrate in x∫₀² (3/2)x² dx = x³/2 evaluated from 0 to 2
- Substitute2³/2 − 0 = 8/2
- The answer4
Copy-ready example
∫₀² ∫₁³ 6x²y dy dx, so x runs 0 to 2 and y runs 1 to 3
A rectangular region
Questions about the Double Integral Calculator
Does the order dy dx or dx dy change the answer?
Not for a continuous function over a sensible region: Fubini's theorem guarantees both orders give 64 for the rectangle above. What changes is the difficulty, and sometimes one order has no elementary antiderivative at all while the other is easy.
What does a double integral actually measure?
If f is a height, it is the volume under the surface and above the region. If f is a density in kg per m², it is the total mass. If f is just 1, the two integrals collapse into the plain area of R.
How do I know which variable to integrate first?
Look at the region. If horizontal slices have simple endpoints, integrate in x first. If vertical slices are simpler, integrate in y first. For the triangle above, vertical slices run from y = 0 up to y = x, so dy goes inside.
Can a double integral come out negative?
Yes, whenever f dips below zero over part of the region. It measures signed volume the same way a single definite integral measures signed area, so subtract-and-cancel behaviour is expected, not an error.