Answer
$$2 \ln 2+3$$
Work Step by Step
Apply cylindrical coordinates. $ x=r \cos \theta ;\\ y= r \sin \theta ;\\ z \gt 0$
We know that $ r(r, \theta)=xi+yj+zk $ or, $ r^2=x^2+y^2+z^2$
Now, $ r_x=\lt 1,0,2x- \dfrac{2}{x} \gt ; \\ r_{\theta}=\lt 0, 1, \sqrt {15} \gt $
Also, $|r_x\times r_{y}|=\dfrac{2}{x}+2x $
Now, $$ Area=\int_1^{2} \int_0^{1} (\dfrac{2}{x}+2x) \space dy \space dx \\=[2 \ln x+x^2]_1^{2} \\= 2 \ln 2+3$$
