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)\(\frac{8}{3} sq.units\)

Correct answer is \(\frac{4}{3}sq.units\)

Explaination::

Given curve is \( y =2 x-x^{2}\) or \((x-1)^{2} =-(y-1)\)

The curve cut the x -axis at (0,0) and (2,0) .

\(\therefore \ Required\ area =\int_{0}^{2} y d x\)

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

\(=4-\frac{8}{3}=\frac{4}{3} sq \ unit\)