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

AFind 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) =


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

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Verified Answer :

LammettHash

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

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

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