Answer
$$2$$
Work Step by Step
$$ g(x,y,z) = y -\ln x =0 \\ \nabla g =\dfrac{-i}{x}+j$$
$\implies |\nabla g|=\sqrt {(xi)^2+(yj)^2+(zk)^2 }=\sqrt {\dfrac{1}{x^2}+1}=\sqrt {\dfrac{x^2+1}{x^2}}$
Now, $$F \cdot n =\dfrac{2 xy}{\sqrt {1+x^2}} d \sigma \\=\sqrt {\dfrac{x^2+1}{x^2}} \ d A \\= \iint_{R} (\dfrac{2 xy}{\sqrt {1+x^2}}) ( \sqrt {\dfrac{x^2+1}{x^2}} ) \ du \ dv \\= \int_{0}^{1} \int_{1}^{e} 2y \ dx \ dz \\=\int_{1}^{e} \int_{0}^{1} (2) \times (\ln x \ dx) \ dz \\= \int_{1}^{e} 2 \ln x \ dx \\=2 [x \ln x -x]_{1}^{e} \\=2$$
