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`