VC4.12

The given curve is $r=\cos t i+\sin t j+tk\,$

Hence its parametric equations are $x=\cos t,y=\sin t,z=t\,$ --(1)

$\int_{C}F\cdot,dr=\int_{C}(zi+xj+yk)\cdot(dxi+dyj+dzk)=\int_{C}(zdx+xdy+ydz)\,$

= $\int_{0}^{2\pi}[z\frac{dx}{dt}+x\frac{dy}{dt}+y\frac{dz}{dt}]\,dt\,$

= $\int_{0}^{2\pi}[t(-\sin t)+\cos t (\cos t)+\sin t]\,dt\,$

= $-\int_{0}^{2\pi}t\sin t\,dt+\frac{1}{2}\int_{0}^{2\pi}(1+\cos 2t)\,dt+\int_{0}^{2\pi}\sin t\,dt\,$

= $-[t(-\cos t)_{0}^{2\pi}+(\sin t)_{0}^{2\pi}]+\frac{1}{2}(2\pi)+[-\cos 2\pi+\cos 0]=2\pi+\pi=3\pi\,$