The area (in sq. units) of the region\left \{ \left ( x,y \right ):y^{2}\geqslant 2x \: and\: x^{2}+y^{2}\leq 4x,x\geqslant 0,y\geqslant 0\right \}is

  • Option 1)

    \pi -\frac{4}{3}

  • Option 2)

    \pi -\frac{8}{3}

  • Option 3)

    \pi -\frac{4\sqrt{2}}{3}

  • Option 4)

    \frac{\pi }{2} -\frac{2\sqrt{2}}{3}

 

Answers (1)

 

Area along x axis -

Let y_{1}= f_{1}(x)\, and \, y_{2}= f_{2}(x) be two curve then area bounded between the curves and the lines

x = a and x = b is

\left | \int_{a}^{b} \Delta y\, dx\right |= \left | \int_{a}^{b}\left ( y_{2}-y_{1} \right ) dx\right |

 

- wherein

Where \Delta y= f_{2}\left ( x \right )-f_{1}(x)

 

 x^2+y^2=4x

and y^2=2x

x^2=2x

\Rightarrow x= 0,2

Area = \int_{0}^{2}\left (\sqrt{4x-x^2}- \sqrt{2x} )dx

= \pi - \frac{8}{3}   , after standard calculation.


Option 1)

\pi -\frac{4}{3}

Incorrect

Option 2)

\pi -\frac{8}{3}

Correct

Option 3)

\pi -\frac{4\sqrt{2}}{3}

Incorrect

Option 4)

\frac{\pi }{2} -\frac{2\sqrt{2}}{3}

Incorrect

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