Double Integrals

We cannot change the order of integration so easily.

Integration of Scalar Fields

Think of this in 3D, where is coming towards you in the -axis. So the integral gives the volume.

Region of Type I

Region of Type II

Double Integrals in Polar Coordinates

We rewrite the integrand by setting so

Triple Integrals

See Coordinate System.

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