Double Integral

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

Definition §

Double Integral

Let $f(x,y)$ be a bivariate function with the domain $D\subseteq {\mathbb{R}}^2$. Then the double integral of $f$ is:

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

• $A$ is the infinitesemal area of a region ($\text{d}x\times \text{d}y$).

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 $c\le y\le d$.

• If $f=1$, then the double integral over a domain $D$ simply returns the area of the domain.
• If $f$ is the density of an object, then the triple integral returns the mass of the object.

Rectangular Domain §

The simplest case of double integrals is when they are evaluated over a rectangular domain. That is, the domain can be expressed as the cartesian product of two intervals:

$$R=[a,b]\times [c,d]$$

where $a,b,c,d$ are constants.

If we have $(x,y)\in R$, then this is equivalent to saying $a\le x\le b$ and $c\le y\le d$, or $D=\{(x,y)|a\le x\le b,c\le y\le d\}$

General Domain §

If the domain, $D$, is given as a relation/function instead of a simple rectangular domain, then we must first classify the axis in which the domain is ‘easier to integrate’ over.

For example:

$$D=\{(x,y)\in {\mathbb{R}}^2:0\le x\le 3,x\le y\le 5\}$$

is not a simple rectangular domain

Vertically Simple §

Used when the domain can be split into vertical strips without ‘gaps’.

The domain is of the form:

$$D=\{(x,y)\in {\mathbb{R}}^2:a\le x\le b,l(x)\le y\le u(x)\}$$

That is, our bounds for $x$ are constant, while the bounds for $y$ are functions of $x$.

%%🖋 Edit in Excalidraw, and the dark exported image%%

Integrate with respect to $y$ first! Our integral is of the form:

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

Horizontally Simple §

Used when the domain can be split into horizontal strips without ‘gaps’.

The domain is of the form:

$$D=\{(x,y)\in {\mathbb{R}}^2:c\le y\le d,l(y)\le x\le r(y)\}$$

That is, our bounds for $y$ are constant, while the bounds for $x$ are functions of $y$.

%%🖋 Edit in Excalidraw, and the dark exported image%%

Integrate with respect to $x$ first! Our integral is of the form:

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

Properties §

Let $f,g$ be continuous bivariate functions. Let $c,d$ be constants.

Linearity §

We can expand out linear combinations of double integrals

$${\iint }_{D}c\cdot f(x,y)+d\cdot g(x,y)\text{d}A=c\cdot {\iint }_{D}f(x,y)\text{d}A+d\cdot {\iint }_{D}g(x,y)\text{d}A$$

Dominance Rule §

Essentially applies ordering (inequalities) to double integrals:

$$\forall (x,y)\in D,f(x,y)\ge g(x,y)⟹{\iint }_{D}f(x,y)\text{d}A\ge {\iint }_{D}g(x,y)\text{d}A$$

Subdivision Rule §

Allows taking the union of two disjoint* domains $D_1$ and $D_2$

* $D_1$ and $D_2$ can share their borders, i.e. their boundary points

$$D=D_1\cup D_2⟹{\iint }_{D}f(x,y)\text{d}A={\iint }_{D_1}f(x,y)\text{d}A+{\iint }_{D_2}f(x,y)\text{d}A$$

Theorems §

Applications §

See Applications Of Double & Triple Integrals