Monday 14 January 2019

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




My Previous Posts






Final words 



Thanks for visiting this website and spending your valuable time to read this post regarding how to find area bounded by two parabolas  .If you liked this post , do share it with your friends to benefit them also ,we shall meet in next post , till then bye and take care......


If you are a mathematician Don't forget to visit my Mathematics You tube channel ,Mathematics Website and Mathematics Facebook Page , whose links are given below





No comments:

Post a Comment

Your valuable suggestions are always acceptable to us for betterment of this website