Discussion Forum

The area of the region bounded by the curve y=2x-x² and X- axis is

Answer:

Explanation:

Given equation of curve is y=2x-x²

 $\Rightarrow$       $x^{2}-2x=-y$

$\Rightarrow$        $  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

 $=\int_{0}^{2} y dx$

  $=\int_{0}^{2} (2x-x^{2}) dx=\left[\frac{2x^{2}}{2}-\frac{x^{3}}{3}\right]_{0}^{2}$

 $=\left[4-\frac{8}{3}-0-0\right]=\frac{4}{3}$  sq. units