The area of the region bounded by $y=2 x-x^{2}$ and the $X$-axis is

We have, $$ \begin{aligned} y &=2 x-x^{2} \\ \Rightarrow \quad y &=-\left(x^{2}-2 x\right) \\ \Rightarrow \quad y &=-\left[(x-1)^{2}-1\right] \\ \Rightarrow \quad y &=-(x-1)^{2}+1 \\ \Rightarrow \quad(x-1)^{2}=-(y-1) \end{aligned} <br><img src="https://cdn-question-pool.getmarks.app/pyq/comedk/--1_EAn7QSjo7Ku4HLYoRDiOoRBGgCkWKNLFBahtcc8.original.fullsize.png" alt="Figure" loading="lazy" decoding="async"><br> $\begin{aligned} \text { Required area } &=\int_{0}^{2} y d x \\ &=\int_{0}^{2}\left(2 x-x^{2}\right) d x \\ &=\left[x^{2}-\frac{x^{3}}{3}\right]_{0}^{2}=4-\frac{8}{3} \\ &=\frac{4}{3} \operatorname{sq} \text { unit } \end{aligned}$