Double Integral

A double integral returns the volume of a Bivariate Function, similar to how a normal integral returns the area of a function.

Definition §

Let $f(x,y)$ be a bivariate function. Then the volume, $V$ of $f$ is equal to :

$$V=\iint f(x,y)\text{d}x\text{d}y=\iint f(x,y)\text{d}A$$

• $A$ is the area of a region ($\text{d}x\text{d}y$). Literally $\text{d}x$ multiplied by $\text{d}y$.

In definite form:

$$V={\iint }_{D}f(x,y)\text{d}A={\int }_c^d{\int }_a^{b}f(x,y)\text{d}x\text{d}y$$

• $D$ is the domain rectangle, and can be represented 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:

$$\int {\int }_{D}f(x,y)\text{d}A=\int \int f(x,y)\text{d}x\text{d}y=\int \int f(x,y)\text{d}y\text{d}x$$

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

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