Question

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

Answer 1

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

$\mathbf f(x,y,z)=\langle x,-4y,-z\rangle\implies\mathbf f(x(t),y(t),z(t))=\langle\sin t,-4\cos t,-t\rangle$

$\displaystyle\int_C\mathbf f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=3\pi/2}\mathbf f(x(t),y(t),z(t))\cdot\mathbf r'(t)\,\mathrm dt$
$=\displaystyle\int_{t=0}^{t=3\pi/2}(\sin t\cos t+4\sin t\cos t-t)\,\mathrm dt$
$=\displaystyle\int_{t=0}^{t=3\pi/2}\left(\frac52\sin2t-t\right)\,\mathrm dt$
$=\dfrac52-\dfrac{9\pi^2}8$