Question

${\int }_{-a}^{a}x|x|d{x}_{=}$

• A. $\frac{a}{3}$
• B. $\frac{a^2}{3}$
• C. $\frac{a^2}{2}$
• D. 0

Answer 1

The correct option is D 0

${\int }_{-a}^{a}x|x|dx$
$={\int }_{-a}^0-x^{2}dx+{\int }_0^a{x}^{2}dx$

$=-{\int }_{-a}^0{x}^{2}dx+{\int }_0^a{x}^{2}dx$

$=-{\left[\frac{x^3}{3}\right]}_{-a}^0+{\left[\frac{x^3}{2}\right]}_0^a$

$=-(0+\frac{a^3}{3})+(\frac{a^3}{3}-0)$

$=-\frac{a^3}{3}+\frac{a^3}{3}=0$