A single signed integral can cancel areas when the curves cross. Split wherever the upper boundary changes. Example: y=x3-3x and y=x x3-3x=x x(x2-4)=0 x=-2,0,2. The cubic is above the line on (-2,0); the line is above the cubic on (0,2). A= _-20(x3-4x), dx+ _02(4x-x3), dx A=4+4=8 units2. The signed integral over [-2,2] is zero, not the total area.

A single signed integral can cancel areas when the curves cross. Split wherever the upper boundary changes.

\(x^3-3x=x\Rightarrow x(x^2-4)=0\Rightarrow x=-2,0,2\).

The cubic is above the line on \((-2,0)\); the line is above the cubic on \((0,2)\).

\(A=\int_{-2}^{0}(x^3-4x)\,\mathrm dx+\int_0^2(4x-x^3)\,\mathrm dx\)

\(A=4+4=8\text{ units}^2\). The signed integral over \([-2,2]\) is zero, not the total area.

When the Curves Cross

When the Curves Cross