The area enclosed between the curves y = x | x | and y = x - | x | is: [2024]

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Solution

Q.
The area enclosed between the curves $y = x$ $| x |$ and $y = x -$ $| x |$ is: [2024]

1 $\frac{8}{3}$

2 $\frac{2}{3}$

3 $\frac{4}{3}$

4 $1$

Ans.
(3)

$y = x$ $| x |$ $= {\begin{matrix} x^{2}, & x \geq 0 \\ - x^{2}, & x < 0 \end{matrix}$

$y = x -$ $| x |$ $= {\begin{matrix} 0, & x \geq 0 \\ 2 x, & x < 0 \end{matrix}$

$∴$ $Required Area = \int_{- 2}^{0} (- x^{2} - 2 x) d x$

$= {[- \frac{x^{3}}{3} - x^{2}]}_{- 2}^{0}$

$= \frac{- 8}{3} + 4 = \frac{4}{3}$

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