A signed area, also called an oriented area or algebraic area, assigns a sign to the area enclosed by an oriented closed
curve. For a piecewise smooth parametrized curve 
, the signed area is the line
integral
 |
(1)
|
or equivalently
![A_s(C)=1/2int_(t_0)^(t_1)[x(t)y^'(t)-y(t)x^'(t)]dt.](/images/equations/SignedArea/NumberedEquation2.svg) |
(2)
|
Reversing the curve orientation reverses the sign,
 |
(3)
|
For an oriented polar curve 
, the corresponding formula is
 |
(4)
|
Equivalently, signed area is the winding-number integral
 |
(5)
|
where 
is the contour
winding number of 
about 
. Thus for a self-intersecting curve,
signed area differs in general from both the sum of the absolute areas of the lobes
and the area of their geometric union. This distinction also records repeated traversals,
even when two parametrized curves have the same image.
