EDIT 1/2/3:
The intuitive understanding of Integral Curvature ( the non-dimensional geometrical parameter measured in steridians for surfaces in $ \mathbb R^3 $) seemed to me elusive. In a simple sphere case where Gauss curvature is constant and a certain area of a spherical cap is covered it is comprehensible as a fraction of full sphere maximum value $4 \pi$.
It is a scalar invariant preserved in isometric mappings, an object of first fundamental form of surface theory. Its innate understanding to the extent other isometrically preserved entities is just not there, imho.
I supposed that if the concept is extended to more prismatic surfaces it may even help in building a mechanics model of Gauss-Bonnet theorem with forces, pressure and bending moments.
Anyhow in order to gain better geometric intuition about what Integral Curvature really is (apart from its definition, the sphere and torus examples ), I tried to setup the following problem with geometric surface less round and more prismatic cylindrical surface.
Whatever may be my motivation of this post stated above can be please ignored as a matter of my opinion, for an answer the following question:
Find meridian profile of a surface of revolution that maximizes volume for given integral curvature $\int K\, dA$.
The functional is
$$\int K\, dA - \lambda_1 \, \int \pi \, r^2 \,dz $$ or $$ \int \dfrac{r^{''}}{r(1+r^{'2})^{2}} \, 2 \pi r \sqrt{1+r^{'2}} dz- \lambda_1 \int \pi r^2 dz $$
Lagrangian with adjusted multiplier $a$ introduced
$$ F= \dfrac{r^{''}}{(1+r^{'2})^{3/2}}- \dfrac{r^2}{a^3} $$
and using Euler-Lagrange equation of second degree
$$ F - r' \dfrac{\partial F}{\partial r'} + r{''} \dfrac{\partial F}{\partial r{''}} = c, $$ where $c$ is an arbitrary constant, which after simplification results in the ODE:
$$r^{'} = \tan \phi,\quad \kappa =\dfrac{ r^{''}} {(1+r^{'2})^{3/2} } $$
$$ a^3 \kappa = \dfrac{r^2+c^2}{5-3 \cos^{2} \phi} $$
Results are plotted below for two cases $ a= \pm 1 , c= 0.5, r_i =0.75, r_i^{'} =0 ;$
Every Lagrange Multiplier is a geometrical property of the sought-after curve or surface upto a multiplicative constant .. ..right? So $a$ is the property parameter here. ( Like in Dido's iso-parametric problem has f ( Area, perimeter, Lagrange Multiplier radius = $\lambda$)=0
However, no intuition gained from the new found looped shapes. There is no feel (for me) of integral curvature to correlate to maximum volume here say between two vertical tangent planes...
I request for comments on the validity of the problem formulation and the result. Any remarks would be highly appreciated.
Continuing the inquiry along same lines to find maximum area for given integral curvature, we get ODE
$$ a^2 \kappa =\frac{(c+ r \cos \phi )}{ 5- 3 \cos^2 \phi}$$
and meridian curves ( $ BC: r_i=1, c= \frac12 $) with varying initial slopes as the following:
