Answer
$1248 \pi$
Work Step by Step
The flux through a surface can be defined only when the surface is orientable.
We know that $\iint_S F \cdot dS=\iint_S F \cdot n dS$
Here, $n$ denotes the unit vector.
Since, $dS=n dA=\pm \dfrac{(r_u \times r_v)}{|r_u \times r_v|}|r_u \times r_v| dA=\pm (r_u \times r_v) dA$
$r_s \times r_v=\sqrt 6 (\cos u j+\sin u k)$
and $dS=\sqrt 6 ( \cos u j+\sin u k) dA$
Here, $F(x,y,z) = 26 (yj+zk)$
and $F(r(u,v))=26 \sqrt 6 (\cos u j+\sin u k)$
Rate of heat inflow through the given surface is given by:
$\iint_S F \cdot dS= \int_{0}^4 \int_{0}^{2 \pi} 26 \sqrt 6 (\cos u j+\sin u k) [\sqrt 6 ( \cos u j+\sin u k) dA] $
$=\int_{0}^4 \int_{0}^{2 \pi} 26(6) (\cos^2 u+\sin^2 u) dA$
Hence, $\iint_S F \cdot dS=(26) \cdot (6) \cdot (4) \cdot (2 \pi)=1248 \pi$
