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.

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