Question

Use Green's Theorem to evaluate (int)2xydx+5x^2dy where D is the region bounded by the circle x^2+y^2=16

Answer 1

I got 256, is this right?

Answer 2

greens theorem being int_{}^{C}Pdx+Qdy = int_{}^{}int_{}^{D}(Q*d/dx-P*d/dy)dA\[\int\limits_{}^{}\]

Answer 3

greens theorem being \[\int\limits_{C}^{}Pdx+Qdy = \int\limits_{}^{}\int\limits_{D}^{}(Q_x-P_y)dA\]

Answer 4

I can guess that \[P=2xy \] and \[Q = 5x^2\] which gave me \[\int\limits_{}^{}\int\limits_{D}^{}(10x-2x)dxdy\]

Answer 5

Right. Now check your calculation of the double integral carefully. I'm not sure 256 is right.

Answer 6

I'd calculate it in polar coordinates. The integrand 8x = 8 r cos(theta), and
dx dy = r dr dtheta. r = 0 --> 4, theta = 0 --> 2pi