Using cylindrical coordinates,
[tex]\displaystyle\int_{x=0}^{x=\sqrt3}\int_{y=0}^{y=\sqrt{3-x^2}}\int_{z=5+2x^2+2y^2}^{z=11}\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]
[tex]\displaystyle=\int_{\theta=0}^{\theta=\pi/2}\int_{r=0}^{r=\sqrt3}\int_{z=5+2r^2}^{z=11}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\frac\pi2\int_{r=0}^{r=\sqrt3}r(11-(5+2r^2))\,\mathrm dr[/tex]
[tex]=\displaystyle\pi\int_{r=0}^{r=\sqrt3}(3r-r^3)\,\mathrm dr[/tex]
[tex]=\dfrac{9\pi}4[/tex]
