Question

The area of the region bounded by $\mathrm{y}=2\mathrm{x}-{\mathrm{x}}^2$ and the $\mathrm{x}$-axis.

• A. $\frac{8}{3}\mathrm{sq}\mathrm{units}$
• B. $\frac{4}{3}\mathrm{sq}\mathrm{units}$
• C. $\frac{7}{3}\mathrm{sq}\mathrm{units}$
• D. $\frac{2}{3}\mathrm{sq}\mathrm{units}$

Answer 1

The correct option is B

$\frac{4}{3}\mathrm{sq}\mathrm{units}$

Explanation for the correct option:

Given, $\mathrm{y}=2\mathrm{x}-{\mathrm{x}}^2$

$\mathrm{y}=2\mathrm{x}-{\mathrm{x}}^2$ cuts $\mathrm{x}$-axis at $\left(2,0\right)$

Points of intersection are $\left(0,0\right)$ and $\left(2,0\right)$

Area,

$$\begin{gathered} ={\int }_0^2\left(2\mathrm{x}-{\mathrm{x}}^2\right)d\mathrm{x} \\ ={\left[{\mathrm{x}}^2-\frac{{\mathrm{x}}^3}{3}\right]}_0^2 \\ =4-\frac{8}{3} \\ =\frac{4}{3} \end{gathered}$$

Hence, option $\left(\mathrm{B}\right)$ is the correct answer.