Question Let C be the boundary of the half-annulus
a^2 < (x^2 + y^2) < b^2 and y>=0
in the x-y plane, traversed in the negative direction what is
∫ (5e^(-7x^2) - y^3) dx + x^3 +cosh^2(5y) dy
My attempt ∫c [(5e^(-7x^2) - y^3) dx + (x^3 + cosh^2(5y)) dy]
= -∫c' [(5e^(-7x^2) - y^3) dx + (x^3 + cosh^2(5y)) dy], where C' has positive orientation
= -∫∫ [(∂/∂x)(x^3 + cosh^2(5y)) - (∂/∂y)(5e^(-7x^2) - y^3)] dA, by Green's Theorem
= -∫∫ 3(x^2 + y^2) dA, over the interior of C (or C')
= -∫(θ = 0 to π) ∫(r = a to b) 3r^2 * (r dr dθ), converting to polar coordinates
= -π * (3/4)r^4 {for r = a to b}
= (3π/4)(a^4 - b^4)
Is this the correct method to answering this question? Can someone explain why green theorem works?
The solution is correct; this problem was designed for an application of Green's theorem. If you are asking why Green's theorem applies here, the answer is: because both components of the vector field have continuous derivatives in the half-annulus domain, including its boundary. If you are asking why Green's theorem is true, my answer is: because we have a proof (see a textbook or Wikipedia). If you want an "intuitive" reason why it's true, I think it's better to think of 2D divergence theorem first: the net amount of substance that flows across the boundary of a region is equal to the amount of substance that is produced inside the region. Green's theorem is like that, but with flow along the boundary instead of across it, and with vortices instead of sources/sinks inside the region. The Wiki page shows the equivalence of Green's theorem and 2D divergence theorem.