Decompose the surface into three components,
![\mathbf r_1,\mathbf r_2,\mathbf r_3](https://tex.z-dn.net/?f=%5Cmathbf%20r_1%2C%5Cmathbf%20r_2%2C%5Cmathbf%20r_3)
, corresponding respectively to the cylindrical region and the top and bottom disks:
![\mathbf r_1(u,v)=\begin{cases}x(u,v)=4\cos u\\y(u,v)=4\sin u\\z(u,v)=v\end{cases}](https://tex.z-dn.net/?f=%5Cmathbf%20r_1%28u%2Cv%29%3D%5Cbegin%7Bcases%7Dx%28u%2Cv%29%3D4%5Ccos%20u%5C%5Cy%28u%2Cv%29%3D4%5Csin%20u%5C%5Cz%28u%2Cv%29%3Dv%5Cend%7Bcases%7D)
where
![0\le u\le2\pi](https://tex.z-dn.net/?f=0%5Cle%20u%5Cle2%5Cpi)
and
![0\le v\le5](https://tex.z-dn.net/?f=0%5Cle%20v%5Cle5)
,
![\mathbf r_2(u,v)=\begin{cases}x(u,v)=u\cos v\\y(u,v)=u\sin v\\z(u,v)=0\end{cases}](https://tex.z-dn.net/?f=%5Cmathbf%20r_2%28u%2Cv%29%3D%5Cbegin%7Bcases%7Dx%28u%2Cv%29%3Du%5Ccos%20v%5C%5Cy%28u%2Cv%29%3Du%5Csin%20v%5C%5Cz%28u%2Cv%29%3D0%5Cend%7Bcases%7D)
where
![0\le u\le4](https://tex.z-dn.net/?f=0%5Cle%20u%5Cle4)
and
![0\le v\le2\pi](https://tex.z-dn.net/?f=0%5Cle%20v%5Cle2%5Cpi)
, and
![\mathbf r_3(u,v)=\begin{cases}x(u,v)=u\cos v\\y(u,v)=u\sin v\\z(u,v)=5\end{cases}](https://tex.z-dn.net/?f=%5Cmathbf%20r_3%28u%2Cv%29%3D%5Cbegin%7Bcases%7Dx%28u%2Cv%29%3Du%5Ccos%20v%5C%5Cy%28u%2Cv%29%3Du%5Csin%20v%5C%5Cz%28u%2Cv%29%3D5%5Cend%7Bcases%7D)
where
![0\le u\le4](https://tex.z-dn.net/?f=0%5Cle%20u%5Cle4)
and
![0\le v\le2\pi](https://tex.z-dn.net/?f=0%5Cle%20v%5Cle2%5Cpi)
.
For the cylinder, we have
![\dfrac{\partial\mathbf r_1}{\partial u}\times\dfrac{\partial\mathbf r_1}{\partial v}=\langle4\cos u,4\sin u,0\rangle\implies\left\|\dfrac{\partial\mathbf r_1}{\partial u}\times\dfrac{\partial\mathbf r_1}{\partial v}\right\|=4](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%5Cmathbf%20r_1%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cmathbf%20r_1%7D%7B%5Cpartial%20v%7D%3D%5Clangle4%5Ccos%20u%2C4%5Csin%20u%2C0%5Crangle%5Cimplies%5Cleft%5C%7C%5Cdfrac%7B%5Cpartial%5Cmathbf%20r_1%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cmathbf%20r_1%7D%7B%5Cpartial%20v%7D%5Cright%5C%7C%3D4)
and the integral over this surface is
![\displaystyle\iint_{\text{cyl}}(x^2+y^2+z^2)\,\mathrm dS=4\int_{v=0}^{v=5}\int_{u=0}^{u=2\pi}((4\cos u)^2+(4\sin u)^2+v^2)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_%7B%5Ctext%7Bcyl%7D%7D%28x%5E2%2By%5E2%2Bz%5E2%29%5C%2C%5Cmathrm%20dS%3D4%5Cint_%7Bv%3D0%7D%5E%7Bv%3D5%7D%5Cint_%7Bu%3D0%7D%5E%7Bu%3D2%5Cpi%7D%28%284%5Ccos%20u%29%5E2%2B%284%5Csin%20u%29%5E2%2Bv%5E2%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![=\displaystyle320\int_{u=0}^{u=2\pi}\mathrm du+8\pi\int_{v=0}^{v=5}v^2\,\mathrm dv](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle320%5Cint_%7Bu%3D0%7D%5E%7Bu%3D2%5Cpi%7D%5Cmathrm%20du%2B8%5Cpi%5Cint_%7Bv%3D0%7D%5E%7Bv%3D5%7Dv%5E2%5C%2C%5Cmathrm%20dv)
![=640\pi+\dfrac83\pi(125)](https://tex.z-dn.net/?f=%3D640%5Cpi%2B%5Cdfrac83%5Cpi%28125%29)
![=\dfrac{2920\pi}3](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B2920%5Cpi%7D3)
Bottom disk:
![\dfrac{\partial\mathbf r_2}{\partial u}\times\dfrac{\partial\mathbf r_2}{\partial v}=\langle0,0,u\rangle\implies\left\|\dfrac{\partial\mathbf r_2}{\partial u}\times\dfrac{\partial\mathbf r_2}{\partial v}\right\|=u](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%5Cmathbf%20r_2%7D%7B%5Cpartial%20u%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cmathbf%0A%20r_2%7D%7B%5Cpartial%20v%7D%3D%5Clangle0%2C0%2Cu%5Crangle%5Cimplies%5Cleft%5C%7C%5Cdfrac%7B%5Cpartial%5Cmathbf%20r_2%7D%7B%5Cpartial%20%0Au%7D%5Ctimes%5Cdfrac%7B%5Cpartial%5Cmathbf%20r_2%7D%7B%5Cpartial%20v%7D%5Cright%5C%7C%3Du)
and the integral over the bottom disk is
![\displaystyle\iint_{z=0}(x^2+y^2+z^2)\,\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=0}^{u=4}u((u\cos v)^2+(u\sin v)^2)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_%7Bz%3D0%7D%28x%5E2%2By%5E2%2Bz%5E2%29%5C%2C%5Cmathrm%20dS%3D%5Cint_%7Bv%3D0%7D%5E%7Bv%3D2%5Cpi%7D%5Cint_%7Bu%3D0%7D%5E%7Bu%3D4%7Du%28%28u%5Ccos%20v%29%5E2%2B%28u%5Csin%20v%29%5E2%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![=\displaystyle2\pi\int_{u=0}^{u=4}u^3\,\mathrm du](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle2%5Cpi%5Cint_%7Bu%3D0%7D%5E%7Bu%3D4%7Du%5E3%5C%2C%5Cmathrm%20du)
![=128\pi](https://tex.z-dn.net/?f=%3D128%5Cpi)
The setup for the integral along the top disk is similar to that for the bottom disk, except that
![z=5](https://tex.z-dn.net/?f=z%3D5)
:
![\displaystyle\iint_{z=5}(x^2+y^2+z^2)\,\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=0}^{u=4}u((u\cos v)^2+(u\sin v)^2+5^2)\,\mathrm du\,\mathrm dv](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Ciint_%7Bz%3D5%7D%28x%5E2%2By%5E2%2Bz%5E2%29%5C%2C%5Cmathrm%20%0AdS%3D%5Cint_%7Bv%3D0%7D%5E%7Bv%3D2%5Cpi%7D%5Cint_%7Bu%3D0%7D%5E%7Bu%3D4%7Du%28%28u%5Ccos%20v%29%5E2%2B%28u%5Csin%20%0Av%29%5E2%2B5%5E2%29%5C%2C%5Cmathrm%20du%5C%2C%5Cmathrm%20dv)
![=\displaystyle2\pi\int_{u=0}^{u=4}(u^3+25u)\,\mathrm du](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle2%5Cpi%5Cint_%7Bu%3D0%7D%5E%7Bu%3D4%7D%28u%5E3%2B25u%29%5C%2C%5Cmathrm%20du)
![=528\pi](https://tex.z-dn.net/?f=%3D528%5Cpi)
Finally, the value of the integral over the entire surface is the sum of the integrals over the component surfaces: