Usually, when you need to evaluate a double integral over a region, the integrand is a function f(x,y). You need to notice that the integrand provided by the problem contains the variable z, 1/(x^2 + y^2 + z^2), hence, under the given conditions, you cannot evaluate the given definite integral.

Supposing that you need to evaluate the following double integral over the rectangular region [-a/2,a/2] yields:

int_(-a/2)^(a/2) int_(-a/2)^(a/2) 1/(sqrt(x^2 + y^2))dxdy

Since the order of integration does not matter, either you willl start to integrate with respect to y, or you will start to integrate with respect to x, the results will coincide.