Question

Evaluate the line integral ∫cf⋅dr, where f(x,y,z)=5xi−4yj+3zk and c is given by the vector function r(t)=⟨sint,cost,t⟩, 0≤t≤3π/2.

Answer 1

WIth the parameterization

$\mathbf r(t)=\langle\sin t,\cos t,t\rangle$

we end up with

$\mathbf f(\mathbf r(t))=\mathbf f(\sin t,\cos t,t)=5\sin t\,\mathbf i-4\cos t\,\mathbf j+3t\,\mathbf k$

We also have

$\mathrm d\mathbf r=\mathbf r'(t)\,\mathrm dt=\langle\cos t,-\sin t,1\rangle\,\mathrm dt$

So in the line integral, we get

$\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=\int_{t=0}^{t=3\pi/2}\mathbf f(\mathbf r(t))\cdot\mathbf r'(t)\,\mathrm dt$

$=\displaystyle\int_{t=0}^{t=3\pi/2}(9\sin t\cos t+3t)\,\mathrm dt=\frac98(3\pi^2+4)$