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

A)\(\frac{2}{3} sq.units\)  

B) \(\frac{4}{3}sq.units\) 

C) \(\frac{5}{3} sq.units\) 

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

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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) .

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

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

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

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