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Area in Polar Coordinates
In rectangular coordinates we can find area under a curve by dividing the
region into an increasing number of vertical strips, approximating the
strips by rectangles and taking a limit. In polar coordinates rectangles are
clumsy to work with and it is better to divide the region into wedges by
using rays
The rays divide the region R into n wedge with areas A1, A2,...,An
and the central angles  ,  ,..., . The area of the entire
region can be written as
A=A1+ A2+,..+An==
If 
is small, and if we assume for simplicity that f( )
is nonnegative, then we can approximate the area Ak of the kth
wedge by the area of a sector with central angle
and radius  where =
is any ray that lies in the wedge, we have the following
If we now increase n in such a way that max  , then the sectors
will become better and better approximation of the wedges and it is
reasonable to expect that the above equation will approach the exact value
of area A that is
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