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allb2916
allb2916
30.11.2019 • 
Mathematics

Avery important application of green’s theorem is that the bounded domain cut out by a simple closed curve c has area given by: where c is traversed in the counterclockwise way. find the area of the of the lemniscate in the first quadrant given by the equation: (x 2 + y 2 ) 2 = xy x ≥ 0, y ≥ 0 (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 1 the origin to the typical point of the curve. therefore y = tx 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 finally, it might be useful to do a google search and see what the lemniscate looks like.)

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