Total geometric area under a graph ($A={\int }_a^b|f(x)|dx$, ${\int }_a^b|f(x)|dx$, $a<b$)

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 ${\int }_a^{b}f(x)dx$.

For $f(x)=x$ on $[-1,1]$, the signed integral is zero, whereas the total geometric area is $2{\int }_0^{1}xdx=1$. To evaluate it by antiderivatives, split at sign changes and reverse the sign on intervals where $f$ is negative.