Line, Surface and Volume Integrals
Show that the area of a region R enclosed by a simple closed curve C is given by A = 1 2 H C (xdy ¡ ydx) = H C xdy = ¡ H C ydx. Hence, calculate the area of the ellipse x = a cos `, y = b sin `. Answer In Green’s theorem, put P = ¡y and Q = x. Then I C (xdy¡ydx) = Z Z R (1+1)dxdy = 2 Z Z R dxdy = 2A Therefore, the area of the region is A ...
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