Question: Let \(\mathrm{F}=\left(a x^{2} y+y^{3}+1\right) \mathrm{i}+\left(2 x^{3}+b x y^{2}+2\right) \mathbf{j}\) be a vector field, where \(a\) and \(b\) are constants.

1. (4 points) Find the values of \(a\) and \(b\) for which \(\mathbf{F}\) is conservative.
2. (4 points) For these values of \(a\) and \(b\), find \(f(x, y)\) such that \(\mathbf{F}=\nabla f\).
3. (4 points) Still using the values of \(a\) and \(b\) from part (a), compute \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) along the curve \(C\) such that \(x=e^{t} \cos t, y=e^{t} \sin t, 0 \leq t \leq \pi\).

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