Find the area of the region enclosed between the circles x2+y2=1 and (x−1)2+y2=1.

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Filo · Accepted Answer

x2+y2=1 is a circle with its center at (0,0) and radius =1 unit.And, (x−1)2+y2=1 is a circle with its center at (1,0) and radius =1 unit.The given equations are,x2+y2=1        ---- ( 1 )(x−1)2+y2=1         ---- ( 2 )Using equations ( 1 ) and ( 2 ), we get1−2x=0x=21​Substituting value of x in ( 1 )we get,y=±23​​So, the circles intersect at A(21​,23​​) and B(21​,2−3​​)Required area = Area AOBCA                         =2(areaAOCA)                         =2(AreaODAO+areaDCAD)=2[∫01/2​1−(x−1)2​dx+∫1/21​1−x2​]=2[2(x−1)1−(x−1)2​​+21​sin−1(x−1)]01/2​+2[2x1−x2​​+21​sin−1x]1/21​=⎣⎡​2(21​−1)1−(21​−1)2​​+21​sin−1(21​−1)⎦⎤​−[2(0−1)1−(0−1)2​​+21​sin−1(0−1)]                                                                               +2⎣⎡​(0+21​sin−11)−221​1−41​​​+21​sin−1(21​)⎦⎤​=2⎣⎡​22−1​×23​​​+21​sin−1(−21​)−0−21​sin−1(−1)⎦⎤​+2⎣⎡​21​sin−11−221​×23​​​−21​sin−121​⎦⎤​=−43​​+(−6π​)−(−2π​)+2π​−43​​−6π​=(−6π​+2π​+2π​−6π​)−2×43​​=(32π​−23​​) sq. units

Question Text	Find the area of the region enclosed between the circles $x^{2} + y^{2} = 1$ and $(x - 1)^{2} + y^{2} = 1$ .
Answer Type	Text solution:1
Upvotes	150

Find the area of the region enclosed between the circles $x^{2} + y^{2} = 1$ and $(x - 1)^{2} + y^{2} = 1$ .

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