Concept preview

What you will be able to do

1. 01Find ALL intersection points of two curves first, by setting them equal and solving completely, before writing any integral.
2. 02Recognise that when curves cross at three or more points the top and bottom curves swap, so the total area must be split at each intersection and summed as $\displaystyle\sum\int\left|\text{top}-\text{bottom}\right|$.
3. 03Explain why a single signed integral across the whole span undercounts (or cancels to the wrong value), and set up the correct piecewise sum with the top curve identified separately on each subinterval.

Begin with the fact that drives everything: an intersection point is exactly where the two curves are equal, and it is also exactly where their vertical gap changes sign. Just to the left of a crossing the difference $f(x)-g(x)$ has one sign; just to the right it has the opposite sign. In geometric terms, whichever curve was higher becomes lower. So each interior intersection is a place where the identity of the "top" curve switches. Between two consecutive intersections, though, the curves never meet, so one of them stays strictly on top for that whole stretch — and there the simple $\left(\text{top}-\text{bottom}\right)$ rule holds.