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when we talk about "definite integral of area" in rectangular or polar coordinates, the "area" is quite clear, at least people do it in this way in general:

rectangular coordinate: area between locus y=f(x) and x axis.

polar coordinate: sector area from original point to locus between start and end angles.

parametric equations use third param, such as t, to describle (x,y).

when we use parametric equations in real world (such as physics), is the "definite integral of area" similar to rectangular coordinate or polar coordinate in general or in most of cases?

any samples or explanations are highly appreciated.

thanks.