Answer
$$\pi$$
Work Step by Step
Here, we have: $ \dfrac{dr}{dt}=-\sin t i +\cos t j+\dfrac{1}{6}k$
The work done can be computed as:
$W=\int_a^b F[r(t)] \dfrac{dr}{dt}(dt)$
or, $=\int_0^{2 \pi } (2\sin t i +3 \cos t j+(\cos t+\sin t) k) \cdot (-\sin t i +\cos t j+\dfrac{1}{6}k) dt$
or, $=\int_0^{2 \pi} 2 \cos^2 t-2 \sin^2 t +\cos^2 t +\dfrac{1}{6} \times \cos t+\dfrac{1}{6} \sin t$
or, $ =\int_0^{2 \pi} 2 \cos 2t+\dfrac{1+\cos 2t}{2}+\dfrac{\cos t}{6}+\dfrac{\sin t}{6} $
or, $=\pi$
