Question

Evaluate the following integral:

\[\int\limits_2^8 \left| x - 5 \right| dx\]

 

Answer 1

\[\int_2^8 \left| x - 5 \right| d x\]
\[\text{We know that}, \left| x - 5 \right| = \begin{cases} - \left( x - 5 \right) &,& 2 \leq x \leq 5\\x - 5&,& 5 < x \leq 8\end{cases}\]
\[ \therefore I = \int_2^8 \left| x - 5 \right| d x\]
\[ \Rightarrow I = \int_2^5 - \left( x - 5 \right) dx + \int_5^8 \left( x - 5 \right) dx\]
\[ \Rightarrow I = - \left[ \frac{x^2}{2} - 5x \right]_2^5 + \left[ \frac{x^2}{2} - 5x \right]_5^8 \]
\[ \Rightarrow I = \frac{- 25}{2} + 25 + 2 - 10 + 32 - 40 - \frac{25}{2} + 25\]
\[ \Rightarrow I = 9\]