Double Integral Calculator

A double integral computes the accumulation of a function $f(x, y)$ over a two-dimensional region $D$:

$\iint_D f(x,y)\,dA$

where $dA$ is the infinitesimal area element. In Cartesian coordinates $dA = dx\,dy$; in polar coordinates $dA = r\,dr\,d\theta$.

Common physical meanings:

• $f(x,y) = 1$ gives the area of $D$.
• $f(x,y) = h(x,y)$ (height function) gives the volume under the surface $z = h(x,y)$ above $D$.
• $f = \rho(x,y)$ (surface density) gives the mass of a thin plate.

The key skills are: choose coordinates, set up bounds, and evaluate as iterated single integrals using Fubini's theorem.

Fubini's Theorem

For a continuous $f$ over a rectangle $D = [a, b] \times [c, d]$:

$\iint_D f\,dA = \int_a^b \int_c^d f(x,y)\,dy\,dx = \int_c^d \int_a^b f(x,y)\,dx\,dy$

Either order works, so pick the one that's easier to integrate.

Type I and Type II Regions

Type I ($y$ bounded by curves of $x$):

$D = \{(x,y) : a \leq x \leq b,\ g_1(x) \leq y \leq g_2(x)\}$

$\iint_D f\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx$

Type II ($x$ bounded by curves of $y$):

$D = \{(x,y) : c \leq y \leq d,\ h_1(y) \leq x \leq h_2(y)\}$

$\iint_D f\,dA = \int_c^d \int_{h_1(y)}^{h_2(y)} f(x,y)\,dx\,dy$

Polar Coordinates

For regions with circular symmetry, use $x = r\cos\theta$, $y = r\sin\theta$, $dA = r\,dr\,d\theta$:

$\iint_D f(x,y)\,dA = \iint_D f(r\cos\theta, r\sin\theta)\,r\,dr\,d\theta$

The factor of $r$ from the Jacobian is essential — forgetting it is the most common error.

When to Switch Order of Integration

If an inner integral becomes intractable (e.g., $\int e^{x^2}\,dx$ has no elementary antiderivative), switching the order of integration often makes the problem solvable. Sketch the region first to find equivalent bounds in the other order.

• Wrong bounds order: Inner bounds may depend on outer variables, but outer bounds must be constants. Reversed = wrong answer.
• Forgetting the polar Jacobian: $dA = r\,dr\,d\theta$, not $dr\,d\theta$.
• Not sketching the region: For non-rectangular $D$, a sketch makes Type I vs Type II obvious.
• Trying to integrate impossible inner functions: If you hit $\int e^{x^2}\,dx$ or similar non-elementary integrand, swap the order before giving up.
• Sign errors with negative integrands: If $f$ changes sign over $D$, the double integral can be zero — this is correct, not a mistake to 'fix'.