Question

Suppose \vec{F}(x,y) = -y \sin(x) \vec{i} + \cos(x) \vec{j}.

Answer 1

The portion of the parabola can be parameterized by $\mathbf r(t)=(t,t^2)$ where $0\le t\le3$.

Now the line integral can be computed as

$\displaystyle\int_C\mathbf F(x(t),y(t))\cdot\mathrm d\mathbf r(t)=\int_0^3(-t^2\sin t,\cos t)\cdot(1,2t)\,\mathrm dt$
$=\displaystyle\int_0^3(-t^2\sin t+2t\cos t)\,\mathrm dt$
$=9\cos3$