After converting to cylindrical coordinates:
The limits of integration that I got are $0\le z \le 1 + r, 0\le r \le 2,0\le \theta \le 2\pi$
After solving the last integral I get is: $(128/15)\int_0^{2\pi} cos\theta\,d\theta$
After solving the integral in the end I get zero. I think the limits of integration for $\theta$ are wrong. Any guidance is appreciated.
The limits $0\leq \theta\leq 2\pi$ give you integration over a complete disk around the origin, including positive and negative $x$ coordinates and positive and negative $y$ coordinates. But the original integral has $x \geq 0$ and $y\geq 0,$ no negative $x$ or $y$ coordinates.
Eliminate all parts of your integral that have negative $x$ or $y$ coordinates, that is, eliminate every $\theta$ such that either $\cos\theta < 0$ or $\sin\theta < 0$. What is left should give you a positive integral.