Question

By using the properties of definite integrals, evaluate the integral: \(\int_{2}^{8} |x-5| \,dx\)

Answer 1

Let  \(I\) =\(\int_{2}^{8} |x-5| \,dx\)

It can be seen that (x−5)≤0 on [2,5] and (x−5)≥0 on [5,8].

\(I\) =\(\int_{2}^{8} -(x-5) \,dx\)+\(\int_{2}^{8} (x-5) \,dx\)                                                   \(\bigg(\int_{a}^{b} f(x) \,dx\) =\(\int_{a}^{c} f(x)\)+\(\int_{c}^{b} f(x)\bigg)\)

=\(-\bigg[\frac{x^2}{2}-5x\bigg]^5_2+\bigg[\frac{x^2}{2}-5x\bigg]^8_5\)

=-\(\bigg[\frac {25}{2}\)-25-2+10\(\bigg]\)+\(\bigg[\)32-40-\(\frac {25}{2}\)+25\(\bigg]\)

=9