A definite integral is signed: portions below an axis contribute negatively. Geometric area is non-negative, so split the interval wherever the curve crosses the relevant axis.
Area measured vertically
\[A=\int_a^b|\mathrm{f}(x)|\,\mathrm{d}x\]
Area measured horizontally
\[A=\int_c^d|\mathrm{f}(y)|\,\mathrm{d}y\]
Modulus graphs
For \(y=|\mathrm{f}(x)|\), keep the portions where \(\mathrm{f}(x)\geq0\) and reflect the portions where \(\mathrm{f}(x)<0\) in the \(x\)-axis.
\[|\mathrm{f}(x)|=\begin{cases}\mathrm{f}(x),&\mathrm{f}(x)\geq0,\\-\mathrm{f}(x),&\mathrm{f}(x)<0.\end{cases}\]
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