Calculate ∫ ∂ F 2 x y d x + x 2 d y + ( 1 + x − z ) d zfor the intersection of z = x 2 + y 2 and 2 x + 2 y + z = 7. Go clockwise with respect to the origin.

Question

Calculate      ∫          ∂      F        2  x  y    d  x  +      x    2      d  y  +  (  1  +  x  −  z  )    d  zfor the intersection of   z  =      x    2    +      y    2   and   2  x  +  2  y  +  z  =  7. Go clockwise with respect to the origin.

Rebekah Zimmerman · Accepted Answer

You should consider the surface S given by the intersection of the cylinder   (  x  +  1      )    2    +  (  y  +  1      )    2    ≤  7  +  2  =  9 with the plane   2  x  +  2  y  +  z  =  7. Then       n    =  −  (  2  ,  2  ,  1  )      /    3 (the ellipse   ∂  S is clockwise oriented) and by Stokes&#039; theorem:                              ∫                      ∂            S                                    F                        d                  s                                    =                  ∬          S                curl         (                  F                )                d                  S                                                  =                  ∬                      (            x            +            1                          )              2                        +            (            y            +            1                          )              2                        ≤            9                          (        0        ,        −        1        ,        0        )        ⋅        (        −        (        2        ,        2        ,        1        )        )                d        x        d        y                                          =                  ∬                      (            x            +            1                          )              2                        +            (            y            +            1                          )              2                        ≤            9                          2                d        x        d        y                                          =        2                Area        (        {        (        x        +        1                  )          2                +        (        y        +        1                  )          2                ≤        9        }        )        =        2        ⋅        9        π        =        18        π        .

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