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Problem 18 Easy Difficulty

Evaluate $$ \displaystyle \int_C (y + \sin x) \, dx + (z^2 + \cos y) \, dy + x^3 \, dz $$ where $ C $ is the curve $ \textbf{r}(t) = \langle \sin t, \cos t, \sin 2t \rangle $, $ 0 \leqslant t \leqslant 2\pi $.
[$\textit{Hint:}$ Observe that $ C $ lies on the surface $ z = 2xy $.]

Answer

$\int_{C}(y+\sin x) d x+\left(z^{2}+\cos y\right) d y+x^{3} d z=\pi$

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University of California, Berkeley
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