We denote n-dimensional Lebesgue measure by $\lambda_n$ and $p(x)=p(x_1,x_2)$ be a polynomial where $x_1 \in \mathbb{R}$ and $x_2 \in \mathbb{R}^{n-1}$. In http://www1.uwindsor.ca/math/sites/uwindsor.ca.math/files/05-03.pdf, it is stated that
$$\lambda_{n}(\{x:p(x)=0\})=\int \lambda_{n-1}\{x_2:p(x_1,x_2)=0)\}dx_1$$.
I don't why the above identity is correct.
In the language of probability I was wondering to know if the following is correct. Suppose that $X,Y$ are two random variables. Then $$P(h(X,Y)\in A)=\int P(h(x,Y)\in A)dx \quad ?$$
The first result is just Fubini's Theorem. Your conjecture is false for two reasons: first requirement for such an equation to hold is that the joint distribution should be a product measure. So you have to assume that $X$ and $Y$ are independent. The second problem is integration on RHS is not w.r.t. Lebesgue measure. Instead if $dx$ you get $dF_(x)$ where $F_X$ is the distribution function of $X$.