Mathematics - Area Under Curves Question with Solution | TestHub

To find the area between the parabola $y=2x-x^2$ and the X-axis, we first determine the points of intersection with the X-axis by setting $y=0$.

$2x-x^2=0$

$x(2-x)=0$

$x=0,x=2$

The required area is given by the definite integral of the curve from $x=0$ to $x=2$.

$\text{Area}={\int }_0^2(2x-x^2)dx$

$\text{Area}={\left[x^2-\frac{x^3}{3}\right]}_0^2$

$\text{Area}=\left(2^2-\frac{2^3}{3}\right)-(0-0)$

$\text{Area}=4-\frac{8}{3}=\frac{4}{3}$

Answer: $\frac{4}{3}$