Find the area of the region bounded by the curves \left ( x-1 \right )^{2}+y^{2}= 1\: \: and\: \: x^{2}+y^{2}= 1, using integration.

 

 

 

 
 
 
 
 

Answers (1)

\left ( x-1 \right )^{2}+y^{2}= 1\: \left ( given \right )                    
\left ( x-1 \right )^{2}+\left ( y-0 \right )^{2}= 1                         
Thus,                                                                 
      center=(1,0)                                               
      radius=1   

x^{2}+y^{2}= 1\: \left ( given \right )
\left ( x-0 \right )^{2}+\left ( y-0 \right )^{2}= 1
Thus
 center=(0,0)
  radius=1
Area required = Area  OACB
First, we find Intersection
points A & B

x^{2}+y^{2}= 1\: ---(1)
\left ( x-1 \right )^{2}+y^{2}= 1\: ---(2)
From equation (1)
x^{2}+y^{2}= 1
y^{2}= 1-x^{2}
put y^{2}= 1-x^{2} in equation (2)
\left ( x-1 \right )^{2}+y^{2}= 1
\left ( x-1 \right )^{2}-x^{2}= 0
x^{2}-2x+1-x^{2}= 0
1= 2x
x= \frac{1}{2}
puting x= \frac{1}{2}  in equation (1)