Find the work done in force field

Question

A unit particle is moved in an anticlockwise manner round a circle with center $(0,0,4)$ and radius 2 in the plane $z=4$ in a force field defined as $F=(xy+z)\boldsymbol{\hat{i}} + (2x+y)\boldsymbol{\hat{j}}+(x+y+z)\boldsymbol{\hat{k}}$. Find the work done. How can I solve it?

Answer 1

The parametrized curve you should use is $\gamma(t) = (2\cos t, 2\sin t, 4)$, with $\gamma'(t) = (-2\sin t, 2\cos t, 0)$ and $t \in [0,2\pi]$. Furthermore ${\bf{F}}(\gamma(t)) = (4\sin t \cos t +4, 4\cos t + 2\sin t, 2\cos t + 2\sin t + 4)$ So, you have to calculate $$\int_{0}^{2\pi} (-8\sin^{2}t\cos t-8\sin t+8\cos^{2}t+4\sin t\cos t) dt$$