Question

Find the area of the region bound by the curves $y=6x-x^2$ and $y=x^2-2x$.

Answer 1

Given equations to the curves are
$y=6x-x^2.........\left(i\right)$
$y=x^2-2x.........\left(ii\right)$
Solving equation $\left(i\right)$ and $\left(ii\right)$ , we get $x=0,4$
Required area $={\int }_0^4(6x-x^2)dx-{\int }_0^4(x^2-2x)dx$
$={\int }_0^4(6x-x^2-x^2+2x)dx$
$={\int }_0^4(8x-2x^2)dx$
$={[4x^2-\frac{2x^3}{3}]}_0^4=(64-\frac{128}{3})=\frac{64}{3}=21\frac{1}{3}$ sq.units