Question

Evaluate the following integral:

\[\int\limits_{- 4}^4 \left| x + 2 \right| dx\]

Answer 1

\[\int_{- 4}^4 \left| x + 2 \right| d x\]
\[We\ know\ that\, \left| x + 2 \right| = \begin{cases} - \left( x + 2 \right) &, &- 4 \leq x \leq - 2 \\x + 2 &, &- 2 < x \leq 4\end{cases}\]
\[ \therefore I = \int_{- 4}^4 \left| x + 2 \right| d x\]
\[ \Rightarrow I = \int_{- 4}^{- 2} - \left( x + 2 \right) d x + \int_{- 2}^4 \left( x + 2 \right) d x\]
\[ \Rightarrow I = \left[ - \frac{x^2}{2} - 2x \right]_{- 4}^{- 2} + \left[ \frac{x^2}{2} + 2x \right]_{- 2}^4 \]
\[ \Rightarrow I = - 2 + 4 - 8 - 8 + 8 + 8 - 2 + 4\]
\[ \Rightarrow I = 20\]