To find the area between two curves that cross, you must split the integral at the crossing point. What does a single integral across the whole interval actually compute?

A. The net signed area, so the parts partly cancel

Past the crossing the integrand changes sign, so the second region subtracts from the first. The integral is doing exactly what it always does, signed area, and the splitting is how you tell it you wanted geometric area instead, the same distinction as displacement against distance.