Answer
$$\dfrac{7 \pi \space \sqrt {10}}{3}$$
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$
We have $ z=1; z=\dfrac{4}{3} \implies z=\dfrac{\sqrt {x^2+y^2}}{3}$
Now, $ r_r= \cos \theta \space i+ \sin \theta \space j+\dfrac{1}{3} \space k\\ r_{\theta}=-r \sin \theta \space i+(r \cos \theta) j $
$|r_r \times r_{\theta}|=\dfrac{r \sqrt {10}}{3}$
Now, $$ Area=\int_0^{2 \pi} \int_3^{4} \dfrac{r \sqrt {10}}{3} \space dr \space d \theta \\=\int_0^{2 \pi} \dfrac{7\sqrt {10}}{6} \space d \theta \\=\dfrac{7 \pi \space \sqrt {10}}{3}$$
