Double and Triple Integral Calculator
Evaluate double and triple integrals with variable limits, in either order or in polar coordinates, with each inner integral worked and checked.
About the Double and Triple Integral Calculator
An iterated integral is worked from the inside out: integrate with respect to the inner variable, treating the others as constants, put in its limits (which may depend on the outer variables), then integrate the result over the next variable.
For a region bounded by curves, choose the order that makes the limits simplest: dy dx for vertical strips, dx dy for horizontal strips. Circles and sectors are easier in polar coordinates, where dA = r dr dθ.
The symbolic engine does the integration; AllCalcHub checks each stage by differentiating, and checks the final value against a numerical integration over the same region. If there's no closed form (for example e^(−x²)), the numerical value is given.
The formula
∬_R f dA = ∫ₐᵇ ∫_{g₁(x)}^{g₂(x)} f(x, y) dy dx; dA = r dr dθ
- Area of R
- ∬_R 1 dA
- Volume of a solid
- ∭ 1 dV
Worked example
∫₀¹ ∫₀ˣ xy dy dx
- Inner: ∫₀ˣ xy dy = x·x²/2 = x³/2
- Outer: ∫₀¹ x³/2 dx = 1/8
Questions people ask
How do I enter a region like 0 ≤ y ≤ √(1 − x²)?
Inner limits: from 0 to sqrt(1 - x^2); outer limits: from 0 to 1, with order dy dx.
Do I include the r in polar coordinates?
No: enter the integrand f(r, θ) and the page multiplies by r for you.
Can it change the order of integration?
Yes. For a double integral it works out the limits in the other order: it cuts the region into strips at its corners, finds the left and right boundary of each strip, and rewrites each boundary curve in the other variable (y = x² becomes x = √y). The new integral is checked by integrating it numerically and comparing. When a strip boundary can't be inverted in closed form, the changed order isn't shown.