I have to solve an integral like this:
$$ d = \iint\limits_R{||x||^{-2/3} dA} $$
And the problem I have is described like this:
*there is a disk region R having an area of 100 m² and in this region we have about 150 clients placed in hexagon grid; So, integrating above noted equitons has to provide us some idea about the relationship between average distance from the center point of the region to the hexagons in this region see picture. I suppose it has also to contain some indicators about basic principles of contracted honeycomb geometry where the centers in hexagons are placed much more sparsely as one moves increasingly far from the central point of the region.
now, it looks very simple but I cannot figure out how I define integral bounds or what is the norm vector ||x|| in such case. Is this a distance in the region or the distance between hexagon centers?
in conclusion, my variable d has to result a value close to 46,91...(I just calculated it backwards because I know that for the circle region with 100m² and 150 Clients the optimal value is 28 hexagons. and for instance for 1500m² and 1000 clients we need 131 hexagons. and in this case, with 1000m² d approximates the value 220) but all my attempts to get this value through integration have failed.
If you want to go deeper look inside this paper:
Thank you in advance!