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Evaluate y dr both directly and using Green s theorem where y is the semicircle...

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Calculus

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Evaluate y dr both directly and using Green s theorem where y is the semicircle in the upper half plane from R to R Solution Evaluate y de directly set y 1 7 2 S y dx Lydx ya y 0 x R cost y R sint dx dt dy 0 dt y dx SS dr dx Rsint dt dy R cost dt TR dy 2 dy 2 dy 7 2 ff dx dy dx dy R 0 dt 0 R 0 t n R t R Now we evaluate fyda this time using Green s theorem In this case P x y y and Q x y 0 y r sin 0 x r cose dx dy rdr d0 0 r R R TR 0 0 T Rsint Rsint dt R th r dr de r dr d0

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