Question: Let \(S\) be the annulus \(1 \leq x^{2}+y^{2} \leq 4 .\) Compute \(\int_{\partial S}\left(x y^{2} d y-x^{2} y d x\right)\), both directly and using Green's theorem. (Here \(\partial S\) denotes the boundary of \(S\), oriented so that \(S\) is always on the left as \(\partial S\) is traversed.)

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