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


HOW TO FIND COMMON AREA OF TWO PARABOLAS
Area under Curves
(x2 /4a)2 =  4ax


     x4/16a2 =  4ax

⇒    x4=  64xa3
    x464xa3= 0

Taking 'x' common 

x(x364a3) = 0
Either  x = 0    or x364a3 = 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 .  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

HOW TO FIND COMMON AREA OF TWO PARABOLAS

Watch this video to remove your doubts if any




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