Finding the Area Between Curves That Intersect at More Than Two Points AB & BC

The integrand $f - g$ is signed, so once the curves cross inside the interval a single integral subtracts one piece from another instead of adding them. For $y = x^{3}$ and $y = x$, solving $x(x^{2} - 1) = 0$ gives three crossings at $-1$, $0$ and $1$; each piece has area $\frac{1}{4}$ for a total of $\frac{1}{2}$, while $\int_{-1}^{1}(x - x^{3})\,dx$ is exactly $0$. The procedure is 8.4's with one step inserted: solve completely, split at every interior root, order the curves separately on each piece by testing a point inside it, and add the pieces.

Taking an absolute value at the end is a different operation from integrating an absolute value, since the first lets the pieces cancel before the sign is removed. The curve $y = x^{3} - x^{2} - 2x$ meets the axis at $-1$, $0$ and $2$, enclosing areas $\frac{5}{12}$ and $\frac{8}{3}$ for a total of $\frac{37}{12}$, while the single integral gives $-\frac{9}{4}$ and its size is still not the area. The number of crossings is usually visible in advance from the degree of $f - g$ or the period of a trigonometric pair, and a repeated root is a touch rather than a crossing, so it needs no split; an unnecessary split is harmless and a missing one is not.