They ask me find the following:
$W$ is the solid bounded by the limited right circular cylinder: $$ x^2+y^2=1$$ and the planes: $$z=0, z=4$$ must calculate: $$\iiint_W z\frac{e^{2x^2+2y^2}}{2}\,dx\,dy\,dz$$ My procedure was as follows, I have in cylindrical coordinates: $$ x = r\cos(\theta ), y = r\sin(\theta ), z = z, r^2=x^2+y^2$$ therefore it poses work well $$\int_0^{2\pi}\int_0^1\int_0^4z\frac{e^{2r^{2}}}{2}r\,dz\,dr\,d\theta$$ but at this point I find problems to develop this integral as I could not develop any method; by parts or change by Fubini. Any advice will be of much help, thanks in advance!
Handling $z$ term is easy as it is just a polynomial.
Handling $\theta$ is easy too.
Now to handle the $r$ term. You miss out an $r$. remember when using cylindrical coordinate. We have to use $r\,dr\,d\theta$ rather than just $dr\,d\theta$. This simplifies the problem.