 Solution

In cylindrical coordinates $r=\sqrt{x^2+y^2}$, the bounded solid enclosed by the two conical sheets and the circular cylinder isIts meridional section consists of two triangular regions; revolving them about the $z$-axis gives the solid. The two conical sheets meet at the origin and meet the circular cylinder in its circles at $z=\pm 1$.

Since $\mathbf{F}=(x,y,z)$, its divergence is $3$. Integrate directly with the cylindrical coordinates volume element $rdzdrd\theta$:The same calculation gives $\mathrm{Vol}(V)=4\pi /3$.