Showing posts with label AREA UNDER CURVES. Show all posts
Showing posts with label AREA UNDER CURVES. Show all posts

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

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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
 ⇒   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

HOW TO FIND COMMON AREA OF TWO PARABOLAS

Watch this video to remove your doubts if any




My Previous Posts
  How to find area bounded by two circles 




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