Question

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

(a) Find a vector parametric equation for the parabola y = x^2 from the origin to the point \left(3,9\right) using t as a parameter.
\vec{r}(t) =


(b) Find the line integral of \vec{F} along the parabola y = x^2 from the origin to \left(3,9\right).
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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`