= Total geometric area under a graph
{title2=$A=\int_a^b|f(x)|\,dx$}
{title2=$\int_a^b|f(x)|\,dx$, $a<b$}
For an <integrable function> $f$ on $[a,b]$, the total geometric <area> between its graph and the horizontal axis is $\int_a^b|f(x)|\,dx$. The <absolute value> makes portions above and below the axis contribute positively, unlike the <signed area under a graph> $\int_a^bf(x)\,dx$.
For $f(x)=x$ on $[-1,1]$, the signed <integral> is zero, whereas the total geometric <area> is $2\int_0^1x\,dx=1$. To evaluate it by <antiderivatives>, split at sign changes and reverse the sign on intervals where $f$ is negative.
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