curl of differential volume element?

Question

I have an equation involving $\nabla \times \left( q \vec v \right) $ , where I call "q" the "charge" and "v" the "velocity". I now want to change this to a current density $ q \vec v \to \vec J dV$. So I then have $ \nabla \times \left( \vec J dV \right)$. The question is, does $ \nabla \times \left( \vec J dV \right) = \left( \nabla \times \vec J \right) dV$ , where $ dV = dxdydz $ is a differential of volume. If not, how does one evaluate $ \iiint \nabla \times \left( \vec J dV \right) $? If I can pull out the $ dV $from the curl how do I show that this is correct mathematically?