HOW TO FIND AREA BOUNDED BY TWO CIRCLES , INTEGRATION OF AREA UNDER CURVE

HOW TO FIND AREA OF TWO CIRCLES INTERSECTING EACH OTHERS,

Here we are going to discuss how to find common area of two circles which are overlapping or intersecting at two points with the help of an example

 Let us consider two circles whose equations are  given below
x² + y² =  1^{2                   .......................(1)}
( x - 1 )² + y² = 1²          .....................(2)

Let us draw these circles in coordinate planes, We can compare these equations with standard form of circle to find  the coordinate of  centre of both the circles are (0,0) and (1,0) respectively and radius of both the circles are 1.

If these two circles intersect with each other then we have to find their point/s of intersection.
To find points of intersection subtracting equation (1) from eq (2) , we get 

    ( X - 1 )² + Y² - X² - Y² =  1^2 - 1² 
⇒( X - 1 )²  - X² =  0
 ⇒X ²  +1 ²  - 2×(1)×(X) - X² =  0

⇒1 - 2X = 0

⇒ X = 1/2 ,
Now to find the values of y put the value of x in equation # 1

(1/2)² + Y² =  1²     
   Y² = 1- (1/4) = 3/4

   Y = 土⇃(3/4)
Therefore two points of intersection are B(1/2 , ⇃(3/4)) , C ( (1/2 , -⇃(3/4))

To understand better the solution of  this problem watch this  video 

Required area = shaded  Area , 

we can divide shaded area into four equal parts , As each parts is symmetrical , Therefore to find shaded area  it is sufficient to find the area of any one of four part and then then multiply it with 4.

Hence  Required area = 4 area OBLO = 4 Area BALB
To avoid tedious calculations choose 2nd part to integrate
I= Required area = 4 Area BALB    ------------- (3)

After simplification , we have

    ALSO READ        HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR     

My previous post HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA

  Final words 

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