I'm not really following. It is not common to take the integral of a single representation - which would be akin to taking the integral of a single wavefunction. Integrating over the interaction between two or more representations is a common way of quickly assessing whether there is nonzero overlap (for example) between two orbitals. You bring up C2v - the z, x, and y oriented p-orbitals have linear bases of A1, B1 and B2 symmetry, respectively. The product of (x,x), (z,z) and (y,y) all yield A1 symmetry, indicating that each orbital overlaps with itself - and therefore a non-vanishing overlap integral. On the other hand, none of the products (x,y), (x,z), or (y,z) yield A1 symmetry, indicating that none of the orbitals overlap with each other - meaning they cannot combine to form new orbitals. This is especially important when determining whether orbitals on different nuclei can overlap to form molecular orbitals. Taking the integral of a single representation doesn't make a whole lot of sense - what is the meaning of an overlap between an orbital and nothing at all?