The area (in square units) bounded by the curves $y=2 x^2$ and $y=\max \{x-[x], x+|x|\}$ in between the lines $x=0$ and $x=2$ is

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Aicharya · Accepted Answer

$ \begin{aligned} & y=\max \cdot\{x-[x], \quad x+|x|\} \\ & \Rightarrow \quad y= \begin{cases}2 x, & x \geq 0 \\ \{x\}, & x < 0\end{cases} \end{aligned} $ Shaded region is required Area $ =\left|\int_0^1\left(2 x-2 x^2\right) d x\right|+\left|\int_1^2\left(2 x^2-2 x\right) d x\right| $

$\begin{aligned} & =\left|\int_0^1 2 x d x-\int_0^1 2 x^2 d x\right|+\left|\int_1^2 2 x^2 d x-\int_1^2 2 x d x\right| \\ & =\left|\left[\frac{2 x^2}{2}\right]_0^1-\left[\frac{2 x^3}{3}\right]_0^1\right|+\left|\left[\frac{2 x^3}{3}\right]_1^2-\left[\frac{2 x^2}{2}\right]_1^2\right| \\ & =\left|1-\frac{2}{3}\right|+\left|\frac{16}{3}-\frac{2}{3}-4+1\right|=\frac{1}{3}+\frac{5}{3}=2 \text { sq. units. }\end{aligned}$

The area (in square units) bounded by the curves $y=2 x^2$ and $y=\max \{x-[x], x+|x|\}$ in between the…

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