HOW TO FIND AREA OF THE CIRCLE WHICH IS INTERIOR TO THE PARABOLA

HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA

Area Under Curves

Let  us write two equations of circle and parabola respectively
4x²+ 4y²  = 9 ------------------- (1) 

and x²      = 4y    -------------------(2)

Reducing (1) to standard form by dividing 4 .we get 
x²+ y²  = (3/2)²   
Ist of all draw figures of both the circle and the parabola in cartesian plane.

As it can be seen from figure both  curves intersect each other at two points say A and A' . 
Next we have to find these two coordinates points of intersection . Solving (1) and (2) to find the values of x and y 
Putting  the value of ' x² ' from (2) in (1) we get 

4(4y)+ 4y² = 9

16y + 4y² - 9 = 0

 4y² - 16y -9 = 0

        

 y = (-16+20)/8  and (-16-20)/8
y = 1/2  and -9/2

So Rejecting the -ve value of y ,because when we put negative value (-9/2) in eq (2) , we shall have two complex values of "x" which are not acceptable.
so only put positive value (1/2) of   'y'  in (2) we get two real values of  'x' such that    x= 土⇃2,
Now we can write coordinate M(⇃2,0) and N (-⇃2,0)

Required Area = Shaded area
                         = 2 × Area OBAO 
Note this step carefully ↑

                   

Multiplying every terms with 2 which is written at  beginning  of the previous line.

Putting the values of upper and lower limits of x

    ALSO READ        HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR     

  Final words 

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