Why integrate over a hemisphere (and not a sphere) to solve the rendering equation?
The form of the rendering equation that uses only the BRDF ($f$ in your example, often called $f_r$) and integrates over one hemisphere does not account for transmission.

When adding in transmission, it's fairly common to add a second integral over the opposite hemisphere, using a different BTDF function (bidirectional transmission distribution function). This is equivalent to an integral over the full sphere of directions with a BSDF function, but since that function would usually have to be defined as a piecewise function, writing it as two integrals can be more straightforward.