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Problem 27 Hard Difficulty

Use a graph of the vector field $ \textbf{F} $ and the curve $ C $ to guess whether the line integral of $ \textbf{F} $ over $ C $ is positive, negative, or zero. Then evaluate the line integral.

$ \textbf{F}(x, y) = (x - y) \, \textbf{i} + xy \, \textbf{j} $,
$ C $ is the arc of the circle $ x^2 + y^2 = 4 $ traversed counter-clockwise from $ (2, 0) $ to $ (0, -2) $

Answer

$3 \pi+\frac{2}{3}$

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Video Transcript

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