This is a strange notation. You have for any $\mathbf F$ vector field   $\int_{\mathcal V}  dV \text{div} \mathbf F  = \oint_{\partial \mathcal V} \mathbf F  \cdot d\mathbf A$ , where $\oint$ is a surface integral and ${\partial \mathcal V}$ is the *closed* boundary surface of the volume ${\mathcal V}$. 

Apply this to $\mathbf {curl H}=\mathbf {J + \dot D}$ and note that $\text{div}\mathbf {curl}=0$, you get $$\oint_{\partial \mathcal V} (\mathbf {J + \dot D})  \cdot d\mathbf A=0$$

The same argument holds for Eq.3 where $\text{div}\mathbf B = 0$, thus 
$$\oint_{\partial \mathcal V} \mathbf {B}  \cdot d\mathbf A=0$$