Question about the angle θ Green's theoremGiven Q ( x , y ) = x ⋅ y 2 + y ln ⁡ ( x + x 2 + y 2 ) and P ( x , y ) = x 2 + y 2 calculate the integral ∫ C P d x + Q d y while C = ( x − 1 ) 2 + ( y − 1 ) 2 = 1why θ should be until 2 π?

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Question about the angle   θ Green&#039;s theoremGiven   Q  (  x  ,  y  )  =  x  ⋅      y    2    +  y  ln  ⁡  (  x  +            x      2        +          y      2        ) and   P  (  x  ,  y  )  =            x      2        +          y      2       calculate the integral       ∫    C    P    d  x  +  Q    d  y while   C  =  (  x  −  1      )    2    +  (  y  −  1      )    2    =  1why   θ should be until   2  π?

Porter Mata · Accepted Answer

Step 1You need to rotate by   2  π to make a complete rotation about the circle.Step 2I assume you were thinking   0  &amp;lt;  θ  &amp;lt;      π    2   made sense because the circle is only located within the first quadrant, but with the equations for x and y that you gave in terms of   r  ,  θ, you should be able to see that the center is (1,1) and you need to make a full rotation to trace the circle around this center. Imagine drawing the unit circle and then translating it 1 unit in the positive x direction and 1 unit in the positive y direction.

iroroPagbublh · Accepted Answer

Step 1If you have   x  =  1  +  r  cos  ⁡  θ and   y  =  1  +  r  sin  ⁡  θ then you need to consider the angle around point (1,1), not around point (0,0).Step 2Since C is the full circle around point (1,1), you need all angles   0  ≤  θ  ≤  2  π.

Given Q(x,y)=x cdot y^2+y ln (x+sqrt{x^2+y^2}) and P(x,y)=sqrt{x^2+y^2} calculate the integral int_C P dx+Q dy while C=(x−1)^2+(y−1)^2=1

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