Mathematics

When two curves intersect multiple times, the function $f(x) - g(x)$ changes sign at each intersection point—it's positive where $f$ is above $g$ and negative where $g$ is above $f$. Integrating across all three intersections lumps these signed areas together, so the negative regions partially cancel the positive ones, giving you a net signed area rather than the total enclosed area. To find the true total area, you must split the integral at each intersection point and take the absolute value of each piece, so you're always adding distances rather than subtracting them.

When two curves intersect multiple times, the function $f(x) - g(x)$ changes sign at each intersection point—it's positive where $f$ is above $g$ and negative where $g$ is above $f$. Integrating across all three intersections lumps these signed areas together, so the negative regions partially cancel the positive ones, giving you a net signed area rather than the total enclosed area. To find the true total area, you must split the integral at each intersection point and take the absolute value of each piece, so you're always adding distances rather than subtracting them.