Calculate area of piecewise closed curve

Hello,

I am trying to find the area enclosed by 4 piecewise smooth curves. As can be seen from the figure,

• BLACK curve is a segment of a circle, C
• ORANGE curve is a segment of an ellipse, E1
• GREEN curve is a line, L
• and RED curve is a segment of another ellipse E2

I know all the parameteric equations for the curves and the parameters of each vertices with respect to the curve equations. For example, the vertice at (-17.14,-18.2) has the parameter of 2.3262, in the range of [0, 2*pi], with respect to C.

To find the area enclosed by these curves, I tried to use brute foce way, that is, to integrate each curve segment equations. It works fine for the curves segment where X - Y values are bijective. Otherwise, the integration of somepart of the curve will cancel out some other part of the curve. An example is the BLACK curve.

To my understand, Green's theorem can solve this. But how do I setup the curves to use Green's theorem? And also, Is there other ways to find the area enclosed by these curves conveniently?

Thanks

Answer 1

[Star Strider]

Experiment with the polyarea function.

[Stephen]

Do you mean that I first use polygons (the curve between two vertices is a line) to approximate the figure, then I use polyarea to find the area?

[Star Strider]

Not exactly. Your closed curve is a polygon (at least as far as polyarea is concerned), so if your connected lines can be expressed as one closed curve, polyarea should work.