Question

Evaluate the following integral:

\[\int\limits_0^4 \left| x - 1 \right| dx\]

Answer 1

\[\int_0^4 \left| x - 1 \right| d x\]

 

\[\text{We know that}, \left| x - 1 \right| = \begin{cases} - \left( x - 1 \right) &,& 0 \leq x \leq 1\\x - 1&,& 1 < x \leq 4\end{cases}\]

 

\[ \therefore I = \int_0^4 \left| x - 1 \right| d x\]

 

\[ \Rightarrow I = \int_0^1 - \left( x - 1 \right) dx + \int_1^4 \left( x - 1 \right) dx\]

 

\[ \Rightarrow I = \left[ - \frac{x^2}{2} + x \right]_0^1 + \left[ \frac{x^2}{2} - x \right]_1^4 \]

 

\[ \Rightarrow I = \frac{- 1}{2} + 1 - 0 + 8 - 4 - \frac{1}{2} + 1\]

 

\[ \Rightarrow I = 5\]