Here is the problem:

Here is my progress:

I have solved the unit normal vector n to be $\displaystyle n = \frac{xi + zk}{3}$ and A dotted with n to be $\displaystyle \frac{5xz}{3}$

So I get to the point where:

$\displaystyle \int{\int{(A dotted with n)dS}} = \frac{1}{3}\int{\int{5xzdS}}$

Now I do not know how to proceed from here. I don't know how to define my upper and lower limits and what dS should be equal to. I have tried to read the book(Schaum's Vector Analysis) but still I cannot seem to get how to approach this, since there has been no example of a horizontally inclined cylinder, only of a vertical one. I have been able to solve a similar problem to this but the cylinder was oriented vertically..

I believe the cylinder in question here has a radius of 3 and runs along the y-axis starting from 0 and ending at 8, and it only exists in the first octant because of its boundaries. Please please help