Question

USE GREEN'S THEOREM PLEASE THANK YOU

3.] Show that the area enclosed by a counterclockwise curve C in the plane is given by Verify the formula works for the triangle with vertices (1,0), (0,2), (-1,0)

PLEASE PLEASE be clear with handwriting, will rate,thank you!

Please show this by applying/using Green's theorem

Answer 1

First Green's theorem states that integral_c Pdx + Qdy = doubleintegral_D (dQ/dx - dp/dy) dA where C is positively oriented a simple closed curve. In the plane. D is the region bounded by C and P and Q having continuous partial derivatives in an region containing D. Derivation Since the Area of D is simply doubleintegral_D dA. We want to choose P and Q such that dQ/dx - dP/dy = 1 and then apply green's theorem. A possible choice is P = 0, Q = x which givens. Integral_c Pdx + Qdy = integral_c Q dy = integral x dy or P = -y. Q = 0 A = integral_c -y dx or finally take P = -y/2, Q = x/2 to yield. A = 1/2 integral_c x dy - y dx \r\n for region gamma_1 y = 0, dy = 0 x = t, dx = 1 1/2 integral x dy - y dx = 0. For region gamma_2 y = -2t dy