Answer
$0$
Work Step by Step
Stokes' Theorem can be defined as: $\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr $
The surface is the part of the cone $x=\sqrt {y^2+z^2} $ for which $0 \leq x \leq 2$ and the boundary of this surface is a circle parallel to the yz plane.
Thus, the parameterization of the boundary can be written as: $C: r(t)=2i+2 \cos t j+2 \sin t k \implies dr=0i-2 \sin t j$
Now, $F(r(t))=[arctan (32 \cos t \sin^2 t) i+8 \cos t j+16 \sin^2t k]$
$\iint_{S} curl F \cdot dS=\iint_{C} F \cdot dr=\int_0^{2 \pi} (arctan (32 \cos t \sin^2 t) i+8 \cos t j+16 \sin^2t k) \cdot (0i-2\sin t j+2 \cos t k) dt
\\\\ =\int_{2 \pi}^{0} (-16 \sin t +32 \sin^2 t) (\cos t dt)$
Consider $\sin t =a$ and $\cos t dt=a$
or, $=\int_0^{0} (-16 at +32 a^2 t)da$
or, $=0$
