Net signed area

The area a definite integral computes — area above the axis counts positive, below counts negative.

A definite integral $\int_a^b f(x)\,dx$ computes net signed area, not total (unsigned) area: regions where $f(x)>0$ contribute POSITIVE area, and regions where $f(x)<0$ contribute NEGATIVE area, and these can cancel. To find the TOTAL area enclosed (ignoring sign), integrate $|f(x)|$ instead, which typically requires splitting the interval at every zero of $f$ .

Examples

$\int_{-1}^{1}x\,dx=0$ — the positive area on $[0,1]$ exactly cancels the negative area on $[-1,0]$ .
$\int_{-1}^{1}|x|\,dx=1$ — the TOTAL area, with no cancellation.