Let $f: \mathbb R^2 \rightarrow \mathbb R$ be a smooth strictly convex function with unique minimum at $0$ such that all level sets $A\_x:=\left\{z ; f(z) \le x \right\}$ are compact. Imagine something like $f(z)=\Vert z \Vert^2.$


Define the integral function


$$F(x):=\int\_{A\_x} g(z) dz$$


where $g$ is as smooth as you like. 


*Question: Is there a way to analytically determine an expression for $F''(x)$?*



Second derivative of integral function
Let $B\_u:=\{z\colon f(z)=u\}$ for $u>u\_0:=\min f$. Let $[0,2\pi)\ni t\mapsto(x\_u(t),y\_u(t))$ be any smooth parametrization of $B\_u$, so that 
$B\_u=\{(x\_u(t),y\_u(t))\colon t\in[0,2\pi)\}$. 


(For instance, one may take $(x\_u(t),y\_u(t))=(\rho\_u(t)\cos t,\rho\_u(t)\sin t)$, where $\rho\_u(t):=f\_t^{-1}(u)$ and $f\_t^{-1}\colon(u\_0,\infty)\to(0,\infty)$ is the function inverse to the function $f\_t\colon(0,\infty)\to(u\_0,\infty)$ defined by the formula $f\_t(r):=f(r\cos t,r\sin t)$.) 


Then 
\begin{equation}
 F'(u)=\int\_0^{2\pi}dt\,\sqrt{x'\_u(t)^2+y'\_u(t)^2}\frac{g(x\_u(t),y\_u(t))}{|(\nabla f)(x\_u(t),y\_u(t))|}, \tag{$\ast$}
\end{equation}
and hence
\begin{equation}
 F''(u)=\int\_0^{2\pi}dt\,\frac d{du}\Big(\sqrt{x'\_u(t)^2+y'\_u(t)^2}\frac{g(x\_u(t),y\_u(t))}{|(\nabla f)(x\_u(t),y\_u(t))|}\Big).
\end{equation}


Here, $dt\,\sqrt{x'\_u(t)^2+y'\_u(t)^2}$ is the infinitesimal length element of the curve $B\_u$, and $\frac{du}{|(\nabla f)(x\_u(t),y\_u(t))|}$ is the infinitesimal distance between the curves $B\_u$ and $B\_{u+du}$ near the point $(x\_u(t),y\_u(t))$. 




---


We can verify formula $(\ast)$ for e.g. $f(x,y)=x^2+y^4$ and $g(x,y)=y^2$, in which case we have the following closed-form expressions: for $u>0$
\begin{equation}
 F(u)=\frac{2 \sqrt{\pi } \Gamma \left(\frac{7}{4}\right)}{3 \Gamma
 \left(\frac{9}{4}\right)}\,u^{5/4} ,\quad
 F'(u)=\frac{5 \sqrt{\pi } \Gamma \left(\frac{7}{4}\right)}{6 \Gamma
 \left(\frac{9}{4}\right)}\,u^{1/4}, 
\end{equation}
and the latter expression coincides with the integral in $(\ast)$, with the parametrization $(x\_u(t),y\_u(t))=(u^{1/2}\cos t,u^{1/4}(\sin t)^{[1/2]})$ of $B\_u$, where $w^{[1/2]}:=|w|^{1/2}\,\text{sign}\, w$. 


