differentiation of a volume Given a function $f(x)$. Then we define $D=\{x\mid f(x)>y\}\subset\mathbb{R}^n$ and $\Gamma=\{x\mid f(x)=y\}$. Now we define $S(y)=\int_D dx$. My question is what is the meaning of $S(y)$? Is it "size" (volume) of the domain $D$? How can I obtain $\frac{dS}{dy}$ and what is its meaning?

No, $S(y)$ is the size of the projection of $D$ onto the $x$ axis. The meaning of the derivative is the speed of increase of $f$ when $f(x) = y.$
