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I got an answer of 0 using the direct way:

Let x = acost, y = asint

F(t) = <a^2, asint>

F'(t) = <0, acost>

\(\displaystyle \int F \cdot dr = \int <a^2, asint> \cdot <0, acost>\)

\(\displaystyle = \int^{pi}_0 a^2sintcost dt\)

\(\displaystyle = \frac{1}{2}a^2sin^2t\)

\(\displaystyle = 0\)

For Green's Theorem, I should get the same number, but I got \(\displaystyle -\frac{4}{3}a^3\) and I have no idea where I went wrong.

\(\displaystyle \int \int (\frac{d}{dx}(y) - \frac{d}{dy}(x^2 + y^2)) dA\)

\(\displaystyle = \int \int -2y dA\)

Then I polarized the coordinates where y = rsin(theta)

\(\displaystyle \int^{\pi}_0 \int^a_0 -2rsin(\theta)r dr d(\theta)\)

\(\displaystyle = \int^{\pi}_0 -\frac{2}{3}r^3sin(\theta)\) 0 to a

\(\displaystyle = \int^{\pi}_0 -\frac{2}{3}a^3sin(\theta) d(\theta)\)

\(\displaystyle = \frac{2}{3}a^3cos(\theta) \) 0 to pi

\(\displaystyle = -\frac{4}{3}a^3\)