Double Integrals in Polar
Sometimes a double integral is easier to evaluate using polar coordinates. This is especially true if the region of integration can be easily defined using a polar equation.
The following formula for converted between rectangular and polar coordinates are needed.
The polar coordinates 
of a point are related to the rectangular coordinates 
by the equations
Change to Polar Coordinates in a Double Integral If 
is continuous on a polar rectangle 
given
we convert from rectangular to polar coordinates in a double integral by writing 
and 
, using the appropriate limits of integration for 
and 
, and replacing 
by 
EXAMPLE 1: Evaluate 
, where is the region in the upper half-plane bounded by the circles 
The region can be described as 
It is the half-ring and in polar coordinates it is given by 
