Explained: "Write the integral as a sum of integrals..."

permaneceerc

Answered question

2021-10-25

Write the integral as a sum of integrals without absolute values and evaluate.

\int_{- 2}^{1} | x | d x

Answer & Explanation

tafzijdeq

Skilled2021-10-26Added 92 answers

Step 1
Given integral:
$\int_{- 2}^{1} | x | d x$
Now,
According to defination:
$| x | = {\begin{cases} x, & x \geq 0 \\ - x, & x < 0 \end{cases}$
The value of |x| in interval -2 to 1:
$| x | = {\begin{cases} x, & 0 \leq x \leq 1 \\ - x, & - 2 \leq x \leq 0 \end{cases}$
Therefore,
$\int_{- 2}^{1} | x | d x = \int_{- 2}^{0} - x d x + \int_{0}^{1} x d x$
Step 2
Now,
Using power rule: $\int x^{a} d x = \frac{x^{a + 1}}{a + 1} + c$
$\int_{- 2}^{1} | x | d x = \int_{- 2}^{0} - x d x + \int_{0}^{1} x d x$
$= - {[\frac{x^{2}}{2}]}_{- 2}^{0} + {[\frac{x^{2}}{2}]}_{0}^{1}$
$= - [\frac{0^{2}}{2} - \frac{{(- 2)}^{2}}{2}] + [\frac{1^{2}}{2} - \frac{0^{2}}{2}]$
$= - [- \frac{4}{2}] + [\frac{1}{2}]$
$= \frac{4}{2} + \frac{1}{2}$
$= \frac{5}{2}$

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