A definite integral \(\displaystyle\int_a^b y\,\mathrm{d}x\) measures the signed region between the curve and the \(x\)-axis: positive above the axis, negative below it. Sketch the region first, then turn each signed integral into a positive area.
Above and below the axis
Above the axis: \(A=\displaystyle\int_a^b y\,\mathrm{d}x\)
Below the axis: \(\displaystyle\int_a^b y\,\mathrm{d}x\) is negative, so \(A=-\displaystyle\int_a^b y\,\mathrm{d}x\)
A region partly above and partly below
Total area \(=\displaystyle\int_0^2 y\,\mathrm{d}x-\displaystyle\int_2^3 y\,\mathrm{d}x\). Evaluate each piece separately; the piece below the axis gives a negative integral, so subtracting it (taking its magnitude) makes the contribution positive.
For total area, split at intercepts where the curve changes sign and add the positive areas. A signed integral may be evaluated across an intercept; a touching zero with no sign change does not require a split.
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