Double Integral Calculator

This double integral calculator evaluates ∬ f(x, y) dA over a rectangle a ≤ x ≤ b, c ≤ y ≤ d for a polynomial f such as x + y or 6x²y. It works out the inner integral, then the outer one, gives the exact answer and the average value of f, and checks the result by integrating in the other order.

Result

Enter values and press Calculate to see the result.

Formula

∬Rf(x,y) dA=∫ab(∫cdf(x,y) dy)dx\iint_R f(x, y)\, dA = \int_a^b \left( \int_c^d f(x, y)\, dy \right) dx

An iterated integral: integrate with respect to y holding x fixed, then integrate the result with respect to x.

∫ab∫cdf dy dx=∫cd∫abf dx dy\int_a^b \int_c^d f\, dy\, dx = \int_c^d \int_a^b f\, dx\, dy

Fubini’s theorem: over a rectangle, both orders give the same answer.

How to use this calculator

  1. Type f(x, y) as a polynomial in x and y, such as x*y^2 or x^2 + 3y (xy also means x × y).
  2. Enter the limits: x from a to b and y from c to d.
  3. Choose the order of integration and press Calculate.

Worked examples

Easy: ∬ (x + y) dA over 0 ≤ x ≤ 1, 0 ≤ y ≤ 2

Inner integral: ∫₀² (x + y) dy = [xy + y²/2] from 0 to 2 = 2x + 2.

Outer integral: ∫₀¹ (2x + 2) dx = [x² + 2x] from 0 to 1 = 3.

The rectangle has area 2, so the average value of f is 3/2.

Medium: ∬ xy² dA, integrating x first

Inner integral: ∫₀² xy² dx = [x²y²/2] from 0 to 2 = 2y².

Outer integral: ∫₁³ 2y² dy = [2y³/3] from 1 to 3 = 18 − 2/3 = 52/3.

Area 4, so the average value is 13/3.

Hard: negative limits

Inner integral: ∫₀² (x² + 3y) dy = [x²y + 3y²/2] from 0 to 2 = 2x² + 6.

Outer integral: ∫₋₁¹ (2x² + 6) dx = [2x³/3 + 6x] from −1 to 1 = 4/3 + 12 = 40/3.

Area 4, so the average value is 10/3.

Common mistakes

  • Treating the outer variable as a variable in the inner integral: hold it constant.
  • Using the wrong limits: in dy dx, the inner limits are the y-limits.
  • Forgetting to subtract the lower limit when it isn’t 0.
  • Mixing up the order: the differential written first (dy in dy dx) is integrated first.

Frequently asked questions

How do you evaluate a double integral?

As two single integrals. Integrate with respect to the inner variable, treating the other as a constant, substitute the inner limits, then integrate the result over the outer limits.

Does the order of integration matter?

Not over a rectangle with a polynomial: Fubini’s theorem gives the same answer either way, and the calculator checks that. For non-rectangular regions the limits change when you switch order.

What does a double integral represent?

For f ≥ 0, the volume under the surface z = f(x, y) above the rectangle. Dividing by the rectangle’s area gives the average value of f.

Which functions and regions are supported?

Polynomials in x and y (total degree up to 12) over rectangles a ≤ x ≤ b, c ≤ y ≤ d. Answers are exact fractions.