HOW TO FIND COMMON AREA OF TWO PARABOLAS ,AREA UNDER TWO PARABOLAS , AREA UNDER CURVES

How to find common area of two parabolas  , Area under two parabolas , Area of region bounded by two parabolas .

Let us consider two parabolas whose equations are given by
y² =  4ax  --------------  (1)
x² =  4ay ----------------  (2)

To check whether these parabolas intersect with each others or not And if they intersect then what is/are their point/s of intersection.

How to find Points of Intersection

To find coordinate of points of intersection ,we have to solve equation (1) and (2)

Consider   eq (2) 

 x² =  4ay 

⇒ y  = x² /4a   ---------------(3)

Putting the value of "y" in equation  (1) ,we get

Area under Curves

(x² /4a)² =  4ax

 ⇒    x⁴/16a² =  4ax

⇒    x⁴=  64xa³

 ⇒   x⁴- 64xa³= 0

Taking 'x' common 

x(x³- 64a³) = 0
Either  x = 0    or x³- 64a³ = 0
⇒ x = 0    or    (x)³- (4a)³ = 0 
⇒ x = 0    or    (x)³- (4a)³ = 0 

⇒ x = 0    or    (x-4a)[ (x)²+ (4a)² + (x)(4a)]   = 0 

⇒ x = 0    or    (x-4a) =  0  or  [ (x)²+ (4a)² + (x)(4a)]   = 0 
⇒ x = 0    or    x =  4a  or   (x)²+ (4a)² + (x)(4a)   = 0 

Since   x²+ 4a.x + 16a²   = 0   have no real  roots ,because its discriminant is negative, therefore this quadratic equation have complex roots. And these roots are rejected .

To find values of y

Now putting both  values of  "x"  in eq (3) i. e . y  = x² /4a   ,we get

1st  put x = 0 

y = 0 / 4a = 0         ⇒ when x = 0 then y = 0

and put x = 4 

y =  (4a)² /(4a)

y =  4a                    ⇒ when x = 4a then y = 4a

Hence two points of intersection of (1) and (2)   O(0,0) and A (4a , 4a) .

Now draw two parabolas using their points of intersections as drawn in given picture.

How to Find Required Area

Now to find the area enclosed between two Parabolas.

Required Area = shaded Area =Area OLAMO - Area ONAMO

Watch this video to remove your doubts if any

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How to find area of the circles which is interior to the parabola

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