Battery light comes on faintly when engine is under load-Corolla

turn off your hazards ---you or someone has pushed the hazard button

Cylinder misfire happens most when the engine is under load. Going up a hill puts a load on the engine, so that's when you misfire and that lights the check engine light. Get it checked out sooner rather than later

No
not if you drive it accordingly
steep hills are best negotiated in low fears with plenty of rpm applied
for example on a steep grade with a manual 1st or 2nd or possibly 3rd and the rpm around 2000 constant will cause it no harm
automatics will go to first automatically so keep the rpm at that mark
engine damage is experienced when the gear selected is too high and the engine is placed under extreme load so down shift to allow the engine to work easily
When going down hill select first and only use the brakes to maintain the 2000 rpm range and allow the engine compression to hold the car speed back
the rule of thumb for experienced drivers is this
you select the same gear to go down that it took to come up

When does it jerk , all speeds or under a steady load ? going up a hill ? Would you consider it a misfire ? do you know what a misfire feels like ? Do you have a check engine light on or flashing ? Does your vehicle have coil on plug ignition ?

it is very possible that you got a bad plug wire check and make sure the plug wire is seating right and staying in place

If your battery light is on...you may have lost your alternator.
The battery light or gauge for that matter is merely an indicator to show if the battery is feeling a charge.
If it don't feel the charge...the light comes on.
I'd have your alternator checked at your local parts house...a lot of them will test it for free.

As for the A/C...when an engine gets under a load, the A/C usually cuts out to give the engine less work like going up a steep hill, or passing another car, or if the alternator quits charging the battery.

Hope this helps...

check fuel pressure if its low then it will cause a miss under load

try making sure all belts are tight jode

Disconected Battery and wait about 5 min if when hooked back up if light comes on again I would look at something got unpluged while that other work was going on!!

Since I bought my van in 2002 the check engine light has come on whenever the engine is under load and I hit 70-75 mph. the stored codes were always a random misfire. The dealer threw everything at it, replaced the fuel injection system twice, the rotor and cap three times, plugs, wires, all kinds of sensors. I would NOT give up and kept bringing it back. They FINALLY fixed it when they discovered that there was "not enough clearance at the head for the valves" (?). The heads were removed, machined and some internal parts were replaced. Finally, I can go up a hill, hit passing gear and the light doesn't come on!