In cylindrical coordinates, the volume is given by the integral
[tex]\displaystyle\int_{\theta=0}^{\theta=2\pi}\int_{z=0}^{z=4}\int_{r=\sqrt{z/4}}^{r=\sqrt z}r\,\mathrm dr\,\mathrm dz\,\mathrm d\theta[/tex]
[tex]=\displaystyle2\pi\int_{z=0}^{z=4}\int_{r=\sqrt{z/4}}^{r=\sqrt z}r\,\mathrm dr\,\mathrm dz[/tex]
[tex]=\displaystyle\pi\int_{z=0}^{z=4}r^2\bigg|_{r=\sqrt{z/4}}^{r=
sqrt z}\,\mathrm dz[/tex]
[tex]=\displaystyle\pi\int_{z=0}^{z=4}\left(z-\dfrac z4\right)\,\mathrm dz[/tex]
[tex]=\displaystyle\dfrac{3\pi}4\int_{z=0}^{z=4}z\,\mathrm dz[/tex]
[tex]=\displaystyle\dfrac{3\pi}8z^2\bigg|_{z=0}^{z=4}[/tex]
[tex]=6\pi[/tex]
