Volume of an evolving set

Question

Given a set $ \Omega \subset \mathrm{R}^n$ with smooth boundary I consider a family of sets $ \Omega(t)$, $t>0$ small, such that $ \partial \Omega(t) = \{ g(x,t): x \in \partial \Omega \}$. Here $g$ is given by $$ g(x,t) = \nu(x)(h_1(x)t+h_2(x)t^2) $$ Here $\nu$ is the normal vectorfield on $ \partial \Omega$ and $h_1(x),h_2(x) \in C^{\infinity}(\partial \Omega)$. Using the formulae for the first and second variation of volume $$ vol(\Omega(t))'= \int_{\partial \Omega}< X_0, \nu> \,d\mu_{\partial \mu} $$ $$ vol(\Omega(t))'= \int_{\partial \Omega} < X_0, \nu>' + < X_0, \nu> div_{\partial \Omega} X \,d\mu_{\partial \mu} $$ where $F:(-\epsilon, \epsilon)\times R^n \to R^n$ is the variation such that $F(t, \Omega)= \Omega(t) $ and $\frac{\partial F}{\partial t}=X_t$ I was able to prove $$ vol(\Omega(t)) = vol(\Omega) + \int_{\partial \Omega} th_1(x)+ \frac{t^2}{2}(h_1(x)+ 2h_2(x)) dx + \mathcal{O}(t^3). $$ But what I am in fact interested in is $$ vol(\Omega(t)\backslash \Omega). $$ I suspect that we have $$ vol(\Omega(t)\backslash \Omega) = \int_{\partial \Omega} (th_1(x)+ \frac{t^2}{2}(h_1(x)+ 2h_2(x)))_+ dx $$ where $(...)_+$ is the positive part. Does anyone have an idea how I could prove this?