Help with line integrals and greens theorem

[TheSaxon]

I get an answer for this problem, but its 0 and i think that's wrong. if someone could please, help that'd be great.

Homework Statement

Find the work using the Line Integral Method:
W = Integral of ( Vector F * dr)

Vector Field: F(x,y) = (xy^2)i + (3yx^2)j

C: semi circular region bounded by x-axis and y = squareroot(4-x^2) where y = squareroot(4-x^2) is greater than 0.

Homework Equations

So where vector is <P,Q>,
Work = Integral of (P dx + Q dy) over the region R.

The Attempt at a Solution

So I first parameterized the curve to get P,Q,dx,dy in terms of a common variable:
x = 2cost, y = 2sint for 0 <= t <= pi which implies dx = -2sint and dy = 2cost

but when I carry out the integration, the limits of integration end up making my answer go to 0 because they are between 0 and pi and I always end up with a sin(t) in the result of the integral.

 

[Kreizhn]

The entire benefit of using Green's Theorem is that you don't need to parameterize.

Note that what you have written there is equivalent to

$$\displaystyle \oint \vec{F} \cdot d\vec{x} = \iint_D \left( \frac{F_2}{dx} - \frac{F_1}{dy} \right) dx dy$$

Thus you can very easily figure out $\frac{F_2}{dx}, \;\; \frac{F_1}{dy}$

Furthermore, $-2<x<2$ and $0<y<\sqrt{4-x^2}$

 

[TheSaxon]

ya, sorry, i don't know how to write in those fancy symbols. Anyway was my answer of 0 correcT?

 

[HallsofIvy]

Why ask for help if you ignore the advice? Do the integration over the half disk and see if you also get 0 there.