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