We estimate the current extinction-corrected H$alpha$ star formation rate (SFR) of the different morphological components that shape galaxies (bulges, bars, and disks). We use a multi-component photometric decomposition based on SDSS imaging to CALIFA Integral Field Spectroscopy datacubes for a sample of 219 galaxies. This analysis reveals an enhancement of the central SFR and specific SFR (sSFR $=$ SFR/$M_{star}$) in barred galaxies. Along the Main Sequence, we find more massive galaxies in total have undergone efficient suppression (quenching) of their star formation, in agreement with many studies. We discover that more massive disks have had their star formation quenched as well. We evaluate which mechanisms might be responsible for this quenching process. The presence of type-2 AGNs plays a role at damping the sSFR in bulges and less efficiently in disks. Also, the decrease in the sSFR of the disk component becomes more noticeable for stellar masses around 10$^{10.5}$ M$_{odot}$; for bulges, it is already present at $sim$10$^{9.5}$ M$_{odot}$. The analysis of the line-of-sight stellar velocity dispersions ($sigma$) for the bulge component and of the corresponding Faber-Jackson relation shows that AGNs tend to have slightly higher $sigma$ values than star-forming galaxies for the same mass. Finally, the impact of environment is evaluated by means of the projected galaxy density, $Sigma$$_{5}$. We find that the SFR of both bulges and disks decreases in intermediate-to-high density environments. This work reflects the potential of combining IFS data with 2D multi-component decompositions to shed light on the processes that regulate the SFR.
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