A simple way to show a formula in area

Area in parametric form

Let $C$ be a parametric curve given by :

If the point $P(x,y)$, as t varies from $a$ to $b$, encircles a loop, the area is:

This formula is rather difficult to understand, here is a simple way to show the result.  Referring to the diagram, if T denotes the signed area of the triangle in yellow, then the   increment of T,                 dT = dS + dA  So that,                   Since                            ,         After some short evaluation we have,                    The result follows by taking integration.

An example           An ellipse is given by the curve :                          The area enclosed by the ellipse is therefore :                          Wow! We get our result easily.

Πηγή: qc.edu