We know that the definite integral $\underset{a}{\overset{b}{\int }}ydx$  gives the area of the region, which is bounded by the curve y =  f (x), the x- axis and the two ordinates x = a, x = b.
Now, consider a closed curve represented by the parametric equations x = f (t), y = $\varphi$ (t) , t being parameter. 
We suppose that the curve does not intersect itself. Suppose that as the parameter ‘t’ increases from a value $t_1$  to the value  $t_2$ , the point P(x, y) describes the curve completely in the counter-clockwise sense. The curve being closed, the point on it corresponding to the value $t_2$ of the parameter is the same as the point corresponding to the value  $t_1$ of the parameter. The area of the region bounded by such a curve is given by the formula 
  $A=\frac{1}{2}\underset{t_1}{\overset{t_2}{\int }}(x\frac{dy}{dt}-y\frac{dx}{dt})dt$
 
The above formula gives the area enclosed by any closed curve what so ever, provided only, that it does not intersect itself; there being no restriction as to the manner in which the curve is situated relative to the coordinate axes.