Fundamental Theorem for Line Integrals

(a) Find a function $\displaystyle f$ such that $\displaystyle F=\nabla f$, and (b) use part (a) to evaluate $\displaystyle \oint_C \vec{F}\cdot d \vec{r}$ along the given curve C.

$\displaystyle F(x,y) = <xy^2, x^2y>$

C: $\displaystyle r(t) = <t + sin(\frac{\pi t}{2}), t + cos(\frac{\pi t}{2})>$

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MY work:

**(a)**

$\displaystyle F(x,y)=<xy^2, x^2y>$

Check if it's conservative or not:

$\displaystyle P=xy^2$ and $\displaystyle Q=x^2y$

$\displaystyle \frac{\partial P}{\partial y} = 2xy$

$\displaystyle \frac{\partial Q}{\partial x} = 2xy$

It is conservative.

$\displaystyle f_x = xy^2$

$\displaystyle f=\frac{1}{2}x^2y^2 +g(y)$

$\displaystyle f_y=x^2y +g'(y) = x^2y$. So $\displaystyle g'(y) = 0$, which means $\displaystyle g(y) = K$

Therefore, $\displaystyle f=\frac{1}{2}xy^2 +K$

Is that correct so far? I'm kind of lost here. I'm confused what I'm supposed to do with the whole $\displaystyle F=\nabla f$ thing.