CalculusMultivariable CalculusCollege

Double Integrals

Integrate over rectangles and simple regions with iterated integrals.

A double integral $\iint_R f(x, y)\,dA$ adds up $f$ over a two-dimensional region $R$ . When $f \ge 0$ it is the volume under the surface $z = f(x, y)$ ; with $f = 1$ it gives the area of $R$ . It is computed as an iterated integral: integrate with respect to one variable, treating the other as constant, then integrate the result.

For a rectangle the order of integration does not matter for continuous functions (Fubini's theorem). For curved regions, the limits of the inner integral depend on the outer variable, and switching the order or changing to polar coordinates can turn a hard integral into an easy one.

Key terms

Iterated integral: Over a rectangle, integrate in either order.
$\iint_R f\,dA = \int_a^b\!\int_c^d f(x,y)\,dy\,dx$

Key ideas

Sketch the region and decide which variable has constant limits.
Inner limits may be functions; outer limits must be constants.
Polar form: $dA = r\,dr\,d\theta$ , ideal for circles.
Average value of $f$ over $R$ is the integral divided by the area of $R$ .

Worked example

Evaluate $\int_0^2 \int_0^1 (x + 2y)\,dy\,dx$ .

Inner, in $y$ : $\int_0^1 (x + 2y)\,dy = \left[xy + y^2\right]_0^1 = x + 1$ .
Outer: $\int_0^2 (x + 1)\,dx = \left[\frac{x^2}{2} + x\right]_0^2$ .
$= 2 + 2 = 4$ .

Answer: $4$

Common mistakes

Putting variable limits on the outer integral.
Forgetting the extra $r$ in polar coordinates.
Integrating in the wrong variable first relative to the limits written.

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.Evaluate $\int_0^3 \int_0^2 1\,dy\,dx$ .
Show answer
$6$
2.Evaluate $\int_0^1 \int_0^1 xy\,dy\,dx$ .
Show answer
$\frac{1}{4}$
3.Find the area of a disc of radius 2 using polar coordinates, to two decimals.
Show answer
$4\pi \approx 12.57$

Key terms

Key ideas

Worked example

Common mistakes

Practice questions

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