Whilst reading several books on QFT, I have come across the derivation of the conserved charges due to the symmetry under spacetime-translations. I can follow the derivations, and have that the conserved charges are

$$P^{\alpha}=\int \mathrm{d}^3xT^{0\alpha}~,$$

where $T^{\mu\nu}$ is the energy-momentum tensor. The time-like component gives straightforwardly that 

$$P^0 = \int\mathrm{d}^3x~T^{00} = \int\mathrm{d}^3x~\mathcal{H}~,$$

and the space-like components give

$$P^k = \int\mathrm{d}^3x~T^{0k} = \int\mathrm{d}^3x~\pi(x)\partial^k\phi(x)~.$$

Which corresponds to the momentum. However, I do not immediately see this. Naively, I would have expected that the momentum would correspond to $\int\mathrm{d}^3x~\pi(x)$, but I would also expect physically that the symmetry due to space-translations would result in conservation of momentum.

In short, I can't seem to connect the above equation for $P^k$, what I would expect the momentum to look like naively. Is there a better way than seeing this than that the momentum is *defined* as the conserved space-like charge due to spacetime translations?