Double Integrals

Suppose is defined on .

The double integral of over is defined as:

if the limit exists.

Properties

If on all points of :

If is the union of non-overlapping regions and :

Change of variables

Suppose that is a transformation whose Jacobian is nonzero and that it maps a region in the − plane onto a region in the −plane. and that is continuous on and that and are type I or type II planar regions. And also that is one-to-one, except perhaps on the boundary of . Then:

Fubini’s Theorem

If is continuous on , then:

More generally, this is true if we assume that f is bounded on R, f is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.

Separable function

If can be factored as then:

Non-recatangular regions

is continuous and lies between two continuous functions of .

Suppose .

Similarily it can be extended when two continuous functions of are given.

In Polar Coordinates

is continuous and lies between two continuous functions of .

Suppose .

Common Shapes

All the common shapes in polar coordinates are explained below. They are defined in . The same version can be adapted for .

Archimedes’ Spiral

Gradually spirals outwards from to .

Circle

Center point is . Radius is .

Cardioid

.

Goes through and .

Shape	When
Cardioid
One-loop Limacon
Inner-loop Limacon	$a \lt b

Lemniscate

Suppose .

Resembles the infinity symbol. Center point is . Width . Height .

Rose

n	Number of petals
odd
even

Definition in Polar Coordinates

Note

Volume of a polar rectangle is given by:

is continuous on a polar region which lies in .