cutting into ceiling joists
I'd like to know what are the ramifications of cutting into a 2x8 basement ceiling joist to allow for the centered installation of suspended-ceiling can lights. There would be four cuts total, two each on two different joists. The cuts would be approx. six feet apart on each joist and the dimensions of each cut would be about 2" inches high by 6" across. This would allow for flush installation with the visible edge of the ceiling tile surface.
I'd like someone to relate any issues involved with compromising the integrity of the joists in this scenario. Alternatively, I could use a 5-inch-high can light instead of the 7-inch and then would have to cut out only about a 1/2-inch-high portion. Would this be a better solution?