Question

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

• A. $\frac{2}{3}$ sq. units
• B. $\frac{4}{3}$ sq. units
• C. $\frac{5}{3}$ sq. units
• D. $\frac{8}{3}$ sq. units

Answer 1

Answer: (B)

Given equation of curve is $y=2x-x^2$
$\Rightarrow x^2-2x=-y$
$\Rightarrow x^2-2x+1=-y+1$
$\Rightarrow (x-1{)}^2=-(y-1)$
This is the equation of parabola having vertex (1,1) and open downward.

The parabola intersect the $X$ -axis, put $y=0,$ we get
$0=2x-x^2$
$\Rightarrow x(2-x)=0$
$\Rightarrow x=0,2$
$\therefore$ Area of bounded region between the curve and $X$ -axis
$=\underset{0}{\overset{2}{\int }}ydx$
$=\underset{0}{\overset{2}{\int }}\left(2x-x^2\right)dx={\left[\frac{2x^2}{2}-\frac{x^3}{3}\right]}_0^2$
$=\left[4-\frac{8}{3}-0-0\right]=\frac{4}{3}$ sq units.