Question

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

Answer 1

$\displaystyle\int_C\mathbf F\cdot\mathrm d\mathbf r=\int_0^{3\pi/2}\langle-5\sin t,\cos t,t\rangle\cdot\langle\cos t,-\sin t,1\rangle\,\mathrm dt$
$=\displaystyle\int_0^{3\pi/2}(-6\sin t\cos t+t)\,\mathrm dt=\frac{9\pi^2}8-3$