Double Integral

Just like an integral returns the area covered by a curve, a double integral returns the volume covered by a surface.

$\int \int f(x,y)dxdy$

And just like how the integral of a domain of $f$ gives it’s length if f is 1 : ${\int }_a^{b}1dx=b-a$, the double integral of 1 over a domain gives the area of that domain:

$${\int }_c^d{\int }_a^{b}1dxdx=d(b-a)-c(b-a)=(d-c)\times (b-a)$$

The domain $D$ can also be written as $[a,b]\times [c,d]$, which is equivalent to $a\le x\le b$ and $c\le y\le d$.

Fubini’s Theorem §

Let f : ${\mathbb{R}}^2\to \mathbb{R}$ be a continuous function over the domain $D=[a,b]\times [c,d]$. Then:

$$\int {\int }_{D}f(x,y)dD=\int \int f(x,y)dxdy=\int \int f(x,y)dydx$$

I.e. the order of integration does not matter.

Fubini’s theorem makes integration a lot easier, in some cases.