Multiple integral vector notation

Question

I was reading a state estimation book and encountered the axiom of total probability written in this form $$\int_{\mathbf{a}}^{\mathbf{b}} p(\mathbf{x}) \,d\mathbf{x}=\int_{a_N}^{b_N}\dots \int_{a_2}^{b_2}\int_{a_1}^{b_1}p(x_1,x_2,\dots ,x_N)dx_1dx_2\dots dx_N$$ where $\mathbf{a} = (a_1,a_2,\dots,a_N)$, $\mathbf{b} = (b_1,b_2,\dots,b_N)$, $\mathbf{x} = (x_1,x_2,\dots,x_N)$. I understand the axiom of total probability, but I'm having trouble understanding how the vector notation on the LHS describes the multiple integral on the RHS.