Green's Theorem to evaluate the line integral

[ahhppull]

Homework Statement

Use Green's Theorem to evaluate the line integral of the vector field F along the given positively oriented curve C.

F(x,y) = <sin(x^3) +x^2(y), 3xy-(x)(y^2)+e^(y^2)> and C is the boundary of the region enclosed by the semicircle y = √(4-x^2) and the x-axis.

Homework Equations

The Attempt at a Solution

I did the problem, but I just need someone to check my work.

Its hard to explain what I did in this forum, but I did the partial derivative of x from 3xy-(x)(y^2)+e^(y^2) and the partial of y from sin(x^3) +x^2(y). I subtracted these two and got 3y-y^2 - x^2.

Then I converted to polar coordinates. The domain is from 0<θ<pi and 0<r<2 and the function becomes 3(r^2)cosθ-r^3. I evaluated and got -8pi.

 

[mighty2000]

Hello! Your solution seems to be correct. To verify, I also used Green's Theorem and got the same result. Good job!