Greens theorem.

Find the integral x^2ydx +xy^3dy using Greens theorem if C is the square from (0,0) to (1,0) to (1,1) to (0,1) to (0,0) travelled postively.

Question

Greens theorem.

Find the integral x^2ydx +xy^3dy using Greens theorem if C is the square from (0,0) to (1,0) to (1,1) to (0,1) to (0,0) travelled postively.

anonymous · Answer

Right now you have the line integral:$$\int\limits_C x^2ydx+xy^3dy$$Let:$$M=x^2y, N=xy^3$$Greens Theorem says:$$\int\limits_C Mdx+Ndy= \int\limits \int\limits_R (N_x-M_y)dxdy$$Where N_x is the partial derivative with respect to x, and M_y is the partial derivative with respect to y.  So calculate the partials, and solve that double integral over your region, which is the unit square.

anonymous · Answer

Thanks, i got that far and tried to integrate it for x from 0 to 1 and for y from 0 to 1- but it didnt get the right result- is that wrong?

anonymous · Answer

What answer did you get?

anonymous · Answer

-1/3

anonymous · Answer

its apparently -1/12

anonymous · Answer

Let me work it out real fast and see what I get.

anonymous · Answer

thanks so much

anonymous · Answer

If i integrate in the order dxdy i get -1/3, dydx i get 1/4....i suppose those multiplied =-1/12 but i dont understand why we'd do that

anonymous · Answer

attachment

anonymous · Answer

Oh my goodness....i just realised what i had done- when i intergrated y^3-x^2 i dropped the y^3 as if i was differentiating. ...very silly.

anonymous · Answer

Thats embarrasing. Thanks so much for taking the time to help! really appreciate it.