The formula this page uses
∫ₐᵇ ∫_{g₁(x)}^{g₂(x)} f dy dx = ∫_c^d ∫_{h₁(y)}^{h₂(y)} f dx dy over the same region
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Iterated Integral Calculator
Solve iterated integral problems with clear steps, notation, and a final check.
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Calculus
Iterated Integral Calculator explained
The short version
- An iterated integral is a stack of ordinary integrals, evaluated from the inside out.
- Reversing the order is allowed, but you must redraw the region and rewrite every limit.
- Sometimes one order is impossible and the other is easy, and reversing is the whole trick.
What each part means
SymbolWhat it means
dy dx — Integration orderThe differential closest to f is done first. dy dx means y first.
g₁(x), g₂(x) — Inner limitsThe bottom and top of a vertical slice. They may contain x.
a, b — Outer limitsThe left and right edges of the whole region. Always constants.
region — The shared regionBoth orders must describe the identical set of points, or the answers will not match.
Show your work: a full example
- An integral that stalls∫₀¹ ∫_y¹ sin(x²) dx dy
- Try the inner integral∫ sin(x²) dx has no elementary antiderivative, so this order is a dead end
- Describe the region from the limits0 ≤ y ≤ 1 and y ≤ x ≤ 1, the triangle with corners (0,0), (1,0) and (1,1)
- Describe the same triangle x-first0 ≤ x ≤ 1 and 0 ≤ y ≤ x
- Rewrite in the new order∫₀¹ ∫₀ˣ sin(x²) dy dx
- Inner integral in y is now trivialsin(x²) · x, because sin(x²) is constant with respect to y
- Outer integral∫₀¹ x sin(x²) dx = −cos(x²)/2 from 0 to 1 = (1 − cos 1)/2 ≈ 0.2298
A second, different case
- A different case: check that both orders agree∬ 6x²y dA over the rectangle 0 ≤ x ≤ 2, 1 ≤ y ≤ 3
- Order 1, inner in y6x² · y²/2 from 1 to 3 = 3x²(9 − 1) = 24x²
- Finish in x∫₀² 24x² dx = 8x³ from 0 to 2 = 64
- Order 2, inner in x6y · x³/3 from 0 to 2 = 2y(8) = 16y
- Finish in y∫₁³ 16y dy = 8y² from 1 to 3 = 72 − 8 = 64
- Compare64 = 64, exactly as Fubini's theorem promises for constant limits
Copy-ready example
∫₀¹ ∫_y¹ sin(x²) dx dy
An integral that stalls
Questions about the Iterated Integral Calculator
When is reversing the order actually necessary?
When the inner antiderivative does not exist in elementary form. sin(x²), e^{x²} and (sin x)/x are the classic three. Reversing often supplies exactly the extra factor of x needed to make a substitution work, as it does above.
Do I always have to sketch the region?
Almost always, yes. The limits alone are easy to misread, and a picture of the triangle makes it obvious that y ≤ x ≤ 1 with 0 ≤ y ≤ 1 is the same set of points as 0 ≤ y ≤ x with 0 ≤ x ≤ 1.
What is Fubini's theorem in plain words?
If the function is continuous on the region, you can slice it in either direction and get the same total. It is why the rectangle example gives 64 both ways, and it is what licenses the whole reversing trick.
Can an iterated integral have three or more layers?
Yes, and the rule does not change: work from the innermost differential outward, and only the outermost limits are constants. Three layers is a triple integral over a solid, and the same reordering logic applies.