Let $f\_1,f\_2$ be two positive functions on $\Omega\_1, \Omega\_2 \subset R^2$ with $f\_1|\_{\partial \Omega\_1}=f\_2|\_{\partial \Omega\_2}=0$. For every $\lambda>0$, denote the the area of the domain enclosed by $f\_{i}=\lambda$ by $A\_i(\lambda)$, $i=1,2$. Assume 


$A\_1(\lambda) \leq A\_2(\lambda)$ for all $\lambda >0$. 


Is 


$\int\_{\Omega\_1}f\_1 dx \leq \int\_{\Omega\_2}f\_2 dx?$



Level sets and integral of functions of two variables
It is not clear what you mean by "enclosed by". If this is just the set of points where $f\_i(x)>\lambda$, then the answer is "yes", since $$\int\_\Omega fdx=\int\_0^\infty A(\lambda)d\lambda.$$ This is just Fubuni's theorem applied to the subgraph of $f$. 


