Perhaps the best way to compute this volume is by converting to cylindrical coordinates.
If the region is denoted \(D\), then the volume is given by
\(V=\displaystyle\iiint_D\mathrm dV\)
\(V=\displaystyle\int_0^4\int_0^{2\pi}\int_{z/4}^zr\,\mathrm dr\,\mathrm d\theta\,\mathrm dz\)
\(V=\displaystyle\pi\int_0^4\left(r^2\bigg|_{r=z/4}^{r=z}\right)\,\mathrm dz\)
\(V=\displaystyle\pi\int_0^4\left(z^2-\dfrac{z^2}{16}\right)\,\mathrm dz\)
\(V=\displaystyle\frac{15\pi}{16}\int_0^4z^2\,\mathrm dz\)
\(V=\displaystyle\frac{5\pi}{16}z^3\bigg|_{z=0}^{z=4}\)
\(V=20\pi\)
