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Leah Pope 2022-06-24 Answered
Calculate
F 2 x y d x + x 2 d y + ( 1 + x z ) d z
for the intersection of z = x 2 + y 2 and 2 x + 2 y + z = 7. Go clockwise with respect to the origin.
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Answers (1)

Rebekah Zimmerman
Answered 2022-06-25 Author has 32 answers
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' 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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