The area between two curves is ∫(top − bottom) dx. Why does that formula work even when both curves dip below the x-axis, where each individual integral would be negative?

C. Because the subtraction measures the vertical gap between the curves.

Test it with f = −1 above g = −3: the individual integrals are −1 and −3, and their difference is 2, the correct area. The integrand is the height of a strip, and where each curve sits relative to the axis never enters. What does need care is curves that cross, since the integrand changes sign, which is why finding the intersections comes first.