HOW TO FIND AREA BOUNDED BY THREE LINES AND CIRCLES , AREA UNDER CURVES BY INTEGRATION METHOD

How to find common area of three lines and one circles which are  intersecting at different  points with the help of an example.

 Given Lines and Curves

Consider one circle and three lines whose equations are  given below
( x - 1 )² + y² = 1^{2            ......................(1)}

      y = x                            .....................(2)

y = -√3 (x-2)                      .....................(3)

y = 0                                   .....................(4)

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

How To Draw Figure

1st of all check whether these lines  intersect with circle or not . And if these lines intersect with each other or with circle then what are their coordinates of points of intersections.

Solve (1) and (2)

Putting the value of  'y' from (2) in equation (1), we get

( x - 1 )² + x² = 1²   

  ⇒  x²  + 1² -2×1×x +x² = 1²   

   ⇒  -2x +x² = 0

    ⇒   2x(-1+x) = 0

Either  2x= 0 or (-1+x) = 0

          ⇒    x = 0 or x = 1

Now putting the values of x in (2) we get 

x = 0 when x = 0  and y = 1 when x = 1

Therefore points of intersection of (1) and (2) are

O( 0 , 0 ) and A( 1, 1 )

Solve (1) and (3)

Putting the value of  'y' from (3) in equation (1), we get

( x - 1 )² + [-√3 (x-2)]² = 1² 

⇒  x ² + 1 ²  - 2x +  3 (x-2)² = 1

 ⇒ x ² + 1- 2x +  3 [ x ² +4 - 4x] =1

  ⇒ x ² + 1- 2x +  3x ² + 12 - 12x -1 = 0 

  ⇒  4x ² - 14x +12 = 0 

   ⇒  2x ² - 7x + 6 = 0 ,    By Factorisation Method

    ⇒ 2x ² - 4x - 3x  + 6 = 0 

  ⇒ 2x(x-2) -3( x - 2) = 0

    ⇒ (x-2)( 2x - 3) = 0

Either (x-2) = 0   or ( 2x - 3) = 0

   x = 2 and x = 3/2

To find the values of y , put both the values of  'x' in (3) .i.e. in  y = -√3 (x-2) 

when x = 2 ,    then  y = 0

and    x = 3/2 , then  y = √3 /2

Therefore points of intersection of (1) and (3) are

C( 2 , 0 ) and B( 3/2, √3 /2 )

Solve (2) and (3)

is the coordinate of Point of intersection of Line (2) and (3)

Same problem with the help of this Video ⇊

How to Find Required Area

Required Area = Shaded Area = Area of ΔOAL + Area of Curve ABMLA+ Area of Δ BCM

After simplification , we get  

My Previous Posts

 How to find area bounded by two circles 

 How to integrate integral with square root in numerator     

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


How to find common area of two parabolas.

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