## 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

x

( x

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

⇒( X

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

x

^{2}^{ }+ y^{2 }= 1^{2 .......................(1)}( x

^{ }- 1 )^{2 }+ y^{2 }= 1^{2 }.....................(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 }+ Y^{2}- X^{2}^{ }- Y^{2 }= 1^{2 }^{-}1^{2 }⇒( X

^{ }- 1 )^{2 }- X^{2}^{ }= 0^{ }⇒X^{2 }+1^{2 }- 2×(1)×(X) - X^{2}^{ }= 0
⇒1 - 2X = 0

⇒ X = 1/2 ,

Now to find the values of y put the value of x in equation # 1

(1/2)Now to find the values of y put the value of x in equation # 1

^{2}^{ }+ Y^{2 }= 1^{2 }^{ }Y^{2 }= 1- (1/4) = 3/4^{ }Y^{ }= åœŸ⇃(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.

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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Kindly publish the solution for this problem using integral x dy.

ReplyDeleteWhat is the problem

DeleteNo problem sir. Just wanted to see the workings when we integrate with y limits.

ReplyDeletewhere did the formula come from?

ReplyDelete