Answer
work done $\int_C \textbf F \cdot \textbf{dr} = -10\pi^2$
Work Step by Step
The given path C can be written as
$\textbf r(t) = 2\cos t \textbf i +2\sin t\textbf j +t \textbf k, 0\le t \le 2\pi\\
= x(t) \textbf i +y(t)\textbf j +z(t) \textbf k$
It follows that $x(t) = 2\cos t, y(t) = 2\sin t, z(t) = t$
$\textbf F(x,y,z) = x \textbf i + y \textbf j -5z \textbf k\\
\Rightarrow \textbf F = 2\cos t \textbf i + 2\sin t \textbf j - 5t \textbf k\\
\textbf r' = -2\sin t \textbf i + 2\cos t\textbf j+ \textbf k$
Work done = $\int_C \textbf F \cdot d\textbf r = \int_{t=0}^{2\pi}\textbf F \cdot \textbf r' dt\\
= \int_{t=0}^{2\pi} (2\cos t \textbf i + 2\sin t \textbf j - 5t \textbf k)\cdot (-2\sin t \textbf i + 2\cos t\textbf j+ \textbf k)dt\\
= \int_0^{2\pi} (-4\cos t.\sin t + 4 \sin t . \cos t -5t)dt\\
= [-\frac{5}{2}t^2]_0^{2\pi} = -10\pi^2$
work done =$ -10\pi^2$ units
