posted by Jennifer .
When we solve for area under a curve, we must consider when the curve is under the axis. We would have to split the integral using the zeros that intersect with the axis.
Would this be for all integrals? What if we just want to "find the integral", without finding it 'in terms of area'?
No, not true for all integrals. But if you are looking for something that only has magnitude, you have to split the integrals, as the area below the axis is NEGATIVE.
On things like vector work (force*dx), the negative would mean work being absorbed, so it might be useful to not divide the integral if you were looking for net work done.