<p>A single signed integral can cancel areas when the curves cross. Split wherever the upper boundary changes.</p>
<p class="paragraph-header">Example: \(y=x^3-3x\) and \(y=x\)</p>
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<p>\(x^3-3x=x\Rightarrow x(x^2-4)=0\Rightarrow x=-2,0,2\).</p>
<p>The cubic is above the line on \((-2,0)\); the line is above the cubic on \((0,2)\).</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(A=\int_{-2}^{0}(x^3-4x)\,\mathrm dx+\int_0^2(4x-x^3)\,\mathrm dx\)</p></div></div>
<p>\(A=4+4=8\text{ units}^2\). The signed integral over \([-2,2]\) is zero, not the total area.</p>

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