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What is a double integral?

A double integral sums a function f(x,y) over a two-dimensional region R, giving, for example, the volume under a surface. For circular regions, switching to polar coordinates can simplify your work.

Double integral formula

The general form is:

Rf(x,y)dA\iint_{R} f(x,y)\,dA

In rectangular coordinates:

y=cd ⁣x=abf(x,y)dxdy\int_{y=c}^{d}\!\int_{x=a}^{b} f(x,y)\,dx\,dy

In polar coordinates (x=rcosθx=r\cos\theta,y=rsinθy=r\sin\theta; Jacobian = rr):

θ=αβ ⁣r=r1r2f(rcosθ,rsinθ)rdrdθ\int_{\theta=\alpha}^{\beta}\!\int_{r=r_{1}}^{r_{2}} f(r\cos\theta,r\sin\theta)\,r\,dr\,d\theta

How to solve a double integral step by step

Step 1: Identify f(x,y) and region R; choose rectangular or polar coordinates.

Step 2: Write down the iterated integral with correct limits.

Step 3: Integrate the inner integral first, then the outer.

Step 4: Simplify and evaluate.

Example: R(x2+y2)dA\iint_{R} (x^2 + y^2)\,dA over the unit disk x2+y21x^2 + y^2 \le 1

Switching to polar coordinates:

x=rcosθy=rsinθdA=rdrdθR(x2+y2)dA=02π ⁣01(r2)rdrdθ=02π ⁣01r3drdθ=[r44]012π=π2 x = r\cos\theta\\ y = r\sin\theta\\ dA = r\,dr\,d\theta\\[6pt] \iint_{R} (x^2 + y^2)\,dA = \int_{0}^{2\pi}\!\int_{0}^{1} (r^2)\,r\,dr\,d\theta = \int_{0}^{2\pi}\!\int_{0}^{1} r^3\,dr\,d\theta = \left[\tfrac{r^4}{4}\right]_0^1 \cdot 2\pi = \tfrac{\pi}{2}

Tips & Tricks

  • Use polar coordinates for circular regions—remember the rr Jacobian.
  • Sketch the region R to determine limits easily.
  • Fubini’s theorem lets you swap dx/dy order if it simplifies your bounds.

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