Interpreting the Signed Area $of$ the Integral

$We$ learned $in$ class that integrals can $be$ used $to$ compute signed area, that $is,$ area that can $be$ positive $or$

negative. For instance, ${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx$ will $be$ negative $iff(x)is$ negative.

For example, say that $we$ wanted $to$ integrate $f(x)=2x\{$:${\displaystyle \underset{-1}{\overset{1}{}}}2xdx).$

$On$ the other hand, $we$ can also integrate ${\displaystyle \underset{a}{\overset{b}{}}}|f(x)|dxif$ all $we$ want $to$ know $is$ total area $(or$ the $un-$

signed area$),$ where area above and below the axis counts $as$ positive either way. $Ifwe$ compute this for the

example function $we$ see that the areas $no$ longer cancel but add together. $Soin$ this case, just using geome$-$

try that the integral ${\displaystyle \underset{-1}{\overset{1}{}}}|2x|dx=2*\frac{1}{2}*1*2=2.$ How can $we$ interpret both the signed and unsigned integral?

For each $of$ the following circumstances, come $up$ with some interpretation $of$ both the signed