Question

A very important application of Green's theorem is that the bounded domain cut out by a simple closed curve C has area given by: x dy - ydx where C is traversed in the counterclockwise way. Find the area of the of folium of Descartes in the first quadrant given by the equation Hint: You will need to directly compute the above line integral and thus you will need a parametrization for the folium in the first quadrant. The parameter to be used is the slope of the line that connects the origin to the typical point of the curve. Therefore ytx and one can plug this into the equation in order to find x and y as expressions of t. Moreover a very useful shortcut is to use y = tx in order to deduce that, at a purely symbolic level, x dy ydx -x dt. Finally, it might be useful to do a Google search and see what the folium looks like.)

Answer 1

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