Suppose $\omega$ is a bounded shape in $\Bbb{R}^3$ and that $\{z : (x,y,z) \in \omega \}=[0,T]$ (that is, the shape is exactly contained in the band $\{z \in [0,T]\}$. If we denote by $\omega_t = \{(x,y,t) \in \omega\}$ the slice of $\omega$ at height $t \in [0,t]$ and $A_t$ the area of $\omega_t$ it is well known that $\int_0^T A_t dt$ gives the volume of $\omega$.

Is there a way to compute $\displaystyle \int_0^T (A_t)^2 dt$ explicitly?

If such a formula exists, it would be useful in an application where I need that the areas of the slices along a certain direction have the smallest variation.

Suppose $\omega$ is a bounded shape in $\Bbb{R}^3$ and that $\{z : (x,y,z) \in \omega \}=[0,T]$ (that is, the shape is exactly contained in the band $\{z \in [0,T]\}$. If we denote by $\omega_t = \{(x,y,t) \in \omega\}$ the slice of $\omega$ at height $t \in [0,T]$ and $A_t$ the area of $\omega_t$ it is well known that $\int_0^T A_t dt$ gives the volume of $\omega$.

Is there a way to compute $\displaystyle \int_0^T (A_t)^2 dt$ explicitly?

If such a formula exists, it would be useful in an application where I need that the areas of the slices along a certain direction have the smallest variation.