Divergence theorem & Kinetic model of plasmas

In one section in my plasma physics notes (see below) on the Vlasov equation, 6D phase space & the kinetic model for plasma, I can’t quite understand how (via Gauss’s divergence theorem) this integral vanishes. This integral appears in the derivation of the conservation of energy for particles in a plasma, using the Vlasov equation.

In particular, I don’t understand why we need to consider the modulus of the velocity going to infinity (which results in the distribution function going to zero) and I would like some insight as to how this affects the (closed) surface integral – it’s quite hard to visualise.

$$
int_{V} mathrm{~d}^{3} x frac{partial}{partial mathbf{x}} cdotleft(mathbf{v} f_{alpha}right)=int_{S} mathbf{v} f cdot mathrm{d} mathbf{S}=0
$$
and this integral is zero because $f$ rapidly tends to zero when $|mathbf{v}| rightarrow infty$.

EDIT: There is likely a typo in the quotation above; $|mathbf{v}| rightarrow infty$ probably should be $|mathbf{x}| rightarrow infty$.

EDIT 2: In addition, it says $frac{partial}{partial mathbf{v}} cdotleft(mathrm{v}^{2}mathbf{a} f_{alpha}right) = 0$ "from Gauss theorem" (i.e. divergence theorem) later on in the notes too.