Upgrade your Vulcan to its maximum level and stock up on ammunition. Throw a few honeypots to lure Mantis into one spot so that he will not attack, then unload on him. This brings his health down significantly.
When he puts up his big shield and starts advancing down the field, go through it. Get behind and follow him, then take out your Judgment or Trident (make sure it is upgraded to at least level 3) and shoot the yellow circle on his back. Use the quick reload tactic and he will go down quick.
The following Bosses in Sector 2 have the corresponding weak spots:
Colossus (Base Boss) Climb the stairs until at the very top and look down at Colossus's head. When the head opens, shoot it with a shotgun for best results (1/5 to 1/4 damage).
Enforcer (City Boss) On the machine's back near the bottom, is a circular lighted area. Your crosshairs will lock on to that area. Shoot it with a shotgun or rocket launcher for best results.
Mantis (Ruins Boss) Get close to the bug and shoot the underside. Shooting its shell will not do anything, as it deflects most of the shots. The best weapon is the Photon MG because it stops Mantis in its tracks.