Visualize it. You are integrating a function P(X) along a thin surface defined by z<f(X)<z+dz. What is volume dV for a given dz? It is a surface dS times dX:

Visualize it. You are integrating a function P(X) along a thin surface defined by z<f(X)<z+dz. What is volume dV for a given dz? It is a surface dS times dX: the amount that X changes for a given dz. Since z=f(x), then dx=dz/|∇f(x)|. So dV=dS*dz/|∇f(x)|