op amp passive sign convention question
We're going to have to start at the beginning.

First, in the US the long bar in a battery is conventionally assigned to be positive, and the short ones negative. This means that when switch position A is connected the input is +1 volt through 125 k. When switch position B is closed the input is -8 volts through 100 k. You have calculated the magnitude of the A current appropriately.

The output of the op amp is properly described by your equations, except that you have the signs wrong. Keep in mind that the op amp is set up in an inverting configuration, so a + input provides a - output, and vice-versa. However, since the input currents in both cases are constant, the integral of the current is not constant. I leave it to you to determine what it really is. This should suggest to you that the op amp will, in fact, reach saturation. 

If you want to be a smart-ass, you might mention on your homework that in general it is not a valid assumption that the saturation voltage of an op amp equals the power supplies, but it's clear that you are expected to assume this.