Question

Sketch the graph y = | x − 5 |. Evaluate $\underset{0}{\overset{1}{\int }}\left|x-5\right|dx$. What does this value of the integral represent on the graph.

Answer 1

We have,
y = | x − 5 | intersect x = 0 and x = 1 at (0, 5) and (1, 4)
Now,
$$\begin{gathered} y=\left|x-5\right| \\ =-\left(x-5\right)\mathrm{For}\mathrm{all}x\in \left(0,1\right) \end{gathered}$$
Integration represents the area enclosed by the graph from x = 0 to x = 1
$$\begin{gathered} A={\int }_0^1\left|y\right|dx \\ ={\int }_0^1\left|x-5\right|dx \\ ={\int }_0^1-\left(x-5\right)dx \\ =-{\int }_0^1\left(x-5\right)dx \\ =-{\left[\frac{x^2}{2}-5x\right]}_0^1 \\ =-\left[\left(\frac{1}{2}-5\right)-\left(0-0\right)\right] \\ =-\left(-\frac{9}{2}\right) \\ =\frac{9}{2}\mathrm{sq}.\mathrm{units} \end{gathered}$$