I was going through David Tong's vectors calculus notes. In Chapter 6.1.3 (Invariant Integrals) he gives the example
Here are some examples. First, suppose that we have a 3d integral over the interior of a sphere of radius $R$, given by
$$T_i = \int_V \rho(r)x_idV=0$$
This must be equal to some invariant 1-tensor (i.e. a vector), but there are no such objects. In other words, we can say immediately that $T_i = 0$. You can check this straightforwardly by doing the integral in, say, spherical polar coordinates.
I can see why this integral would give 0 for a sphere located at the origin, but in the derivation of the general statement, there is not conditions related to the volume that the integral is taken over. So shouldn't it give 0 for any arbitrary volume. If yes, how?
(My apologizes for the poor wording of the question, I'm not very familiar with tensor terminology.)
