The integral of a function over some region measures the total value of the function in that region:
$$T(u)=\int u\thinspace\mathrm{d}V$$
The integral of the squared norm of the gradient of the function measures its variability over some region and is known as the Dirichlet energy:
$$E(u)=\int \lvert\nabla u\rvert^2\thinspace\mathrm{d}V$$
Is there a geometric significance to the integral of the determinant of the Hessian matrix (the square matrix of second-order partial derivatives) of a function over some region?
$$\int\det H(u)\thinspace\mathrm{d}V$$
What other integral functionals of a similar form provide interesting information about the behavior of a function over some region?
The Hessian determinant allows a considerable flexibility to the function, more than the Dirichlet integral. For example, all functions of the form $u(x,y) = \epsilon x^2+\epsilon^{-1}y^2$ have the same Hessian determinant $4$, while taking extremely large values on the unit disk if $\epsilon$ is very large or very small.
The Hessian determinant is the Jacobian determinant of the gradient map $\nabla u$. The geometric significance of the integral of Jacobian is that it's the volume of the image, accounting for multiplicity and orientation. This is also not exactly geometric unless the map happens to be orientation preserving and (more or less) injective. An important case when $\nabla u$ has this property is when $u$ is convex.
For a convex function, the integral $\int_U \det D^2u$ is the volume of the image of $U$ under $\nabla u$. This can be interpreted as some measure of the spread of directions of tangent plane to the graph of $u$ on the boundary of $U$.
In these brief notes Caffarelli explains how this simple fact can be used: for a convex function the supremum of $\nabla u$ is controlled by the oscillation of $u$ on a larger domain $V\Supset U$. Hence, $\int_U \det D^2u\le C (\operatorname{osc}_V u)^n$, which is neat: second derivatives controlled by the function itself.