You would just integrate f and g separately over the appropriate intervals, then add and subtract as needed to get the desired area. This is not, of course, the usual way to find this area, but is perfectly valid.
You haven't shown us the problem, so I don't know for sure what f is, but A is just the negative of the integral of f over some interval. That's how you find the area between a graph below the axis and the axis, because the integral is the negative of the area.
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