If we were to calculate

we would get an answer of

The negative might surprise you.

However, the integral ${\int }_a^b{f}^{\prime }(x)dx$ represents the total change in $y=f(x)$ as $x$ goes from $a$ to $b,$ and so if $f(x)$ is decreasing it makes sense that the total change should be negative.

If the question were asked differently, e.g. find the finite area bound between the curve $y=x^2-1$ and the $x$ axis, then we would give our answer as $|-\frac{4}{3}|=\frac{4}{3}$. This will be made more clear in the next example.

Part of the curve

is sketched below:

Find the shaded area.