Green's Theorem

Question

Verify Green's Theorem for integrate C [(xy + y^2) dx + x^2 dy ] , where C is the closed curve of the region bounded by y=x and y=x^4

Answer 1

So you can evaluate the line integral the way you normally would and get that answer, either parametrize the curve (it would have to be split up into two) or since it's in the form it's in plug in y in terms of x for the first integral and then x in terms of y for the 2nd and use proper bounds... still has to be split up... either way it's a lot of work so you'll see why Green's Theorem, which is really just stokes theorem in 2D, is so great

Then for Green's theorem Suppose that ∫ Pdx + Qdy = ∫∫ (dQ/dx - dP/dy )*dA so integrate (2x-x-2y)*dydx with dy going from x^4 up to x, and x going from 0 to 1... it's just a simple double integral that's not great to solve but it's still way better than doing it the traditional way

Unless I made a mistake, i got 7/18 as the answer, I don't have my work on here because I can't post photos here, but hopefully I lead you in the right direction!

btw please comment or private message if you want calc 3 tutoring! I love multivariable calc and would love to tutor... my rates are very negotiable