Question

${\int }_{-1}^2{\left|x\right|}^{3}dx=$

• A. $\frac{5}{4}$
• B. $\frac{17}{4}$
• C. $\frac{15}{4}$
• D. $\frac{4}{5}$

Answer 1

The correct option is B

$\frac{17}{4}$

Explanation for correct option(s)

Finding the value for the given equation:

Consider the given equation as,

$I={\int }_{-1}^2{\left|x\right|}^{3}dx$

We know that,

$\left|x\right|=\left\{\begin{array}{ll}x,& x\ge 0\\ -x,& x\le 0\end{array}\right.$

Integrating the given equation

$$\begin{gathered} I=-{\int }_{-1}^0{x}^{3}dx+{\int }_0^2{x}^{3}dx \\ =-{\left[\frac{x^4}{4}\right]}_{-1}^0+{\left[\frac{x^4}{4}\right]}_0^2 \\ =-\left[\frac{0^4}{4}-\frac{-1^4}{4}\right]+\left[\frac{2^4}{4}-\frac{0^4}{4}\right] \\ =-\left(0-\frac{1}{4}\right)+\left(\frac{16}{4}-0\right) \\ =\frac{1}{4}+4 \\ I=\frac{17}{4} \end{gathered}$$

Hence, $I={\int }_{-1}^2{\left|x\right|}^{3}dx=\frac{17}{4}$

Therefore, the correct answer is Option (B).