The integral of a function $f\left(x\right)$ between the points $a$ and $b$ is denoted by
${\int}_{a}^{b}f\left(x\right)\mathrm{dx}$
and can be roughly described as the area below the graph of $y\=f\left(x\right)$ and above the $x$-axis, minus any area above the graph and below the $x$-axis, and all taken between the points $a$ and $b$.
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The integral is important because it is an antiderivative for the original function, that is, if
$g\left(t\right)\={\int}_{a}^{t}f\left(x\right)\mathrm{dx}$
then
$g\'\left(x\right)\=f\left(x\right)$.