The parabolas y^{2}=4x\; and\; x^{2}=4y  divide the square region bounded by the lines x=4,y=4 and the coordinate axes. If S_{1},S_{2},S_{3} are respectively the areas of these parts numbered from top to bottom then S_{1}:S_{2}:S_{3}  is

  • Option 1)

    1:2:3

  • Option 2)

    1:2:1

  • Option 3)

    1:1:1

  • Option 4)

    2:1:2

 

Answers (1)

As we learnt in 

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)

 

S_2=\int_{0}^{4}\left (\sqrt{4x}-\frac{x^2}{4} \right )dx

       = \left [ \frac{2x^{3/2}}{3/2} \right ]_{0}^{4}- \left [ \frac{x^3}{12} \right ]_{0}^{4}

       = \frac{4}{3}\times (8-0)- \frac{4^3}{12}

      = \frac{32}{3}- \frac{64}{12}

= \frac{16}{3}

Total area = S+ S+ S= 16

Also S= S3

Thus

  S_1=S_2=S_3=\frac{16}{3}


Option 1)

1:2:3

This is incorrect option

Option 2)

1:2:1

This is incorrect option

Option 3)

1:1:1

This is correct option

Option 4)

2:1:2

This is incorrect option