# Double Integrals

We cannot change the order of integration so easily.

### Integration of Scalar Fields

Think of this in 3D, where $f(x,y)$ is coming towards you in the $z$-axis. So the integral gives the volume.

##### Region of Type I

$∫_{R}f(x,y)dA=∫_{a}∫_{g(x)}f(x,y)dydx$

##### Region of Type II

$∫_{R}f(x,y)dA=∫_{c}∫_{g(y)}f(x,y)dxdy$

## Double Integrals in Polar Coordinates

We rewrite the integrand by setting $x=ρcosϕandy=ρsinϕ$ so $f(x,y)=(ρcosϕ,ρsinϕ)$

$∬_{R_{xy}}f(x,y)dxdy=∬_{R_{ρϕ}}f(ρcosϕ,ρsinϕ)ρdρdϕ$

### Triple Integrals

See Coordinate System.

In general, we double and triple integrals to calculate areas and volumes. The general notation is given by

$Area=∬_{R}dA$ $Volume=∭_{D}dV$