Calculus 3 Quiz
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A student parametrizes the curve of intersection of x2+y2=1x^2 + y^2 = 1 and z=xyz = xy as r(t)=⟨cos⁡t,sin⁡t,cos⁡tsin⁡t⟩\mathbf{r}(t) = \langle \cos t, \sin t, \cos t \sin t \rangle for t∈[0,2π]t \in [0, 2\pi]. When computing ∫Cz ds\int_C z \, ds, they use ds=∣r′(t)∣dtds = |\mathbf{r}'(t)| dt and find ∣r′(t)∣=sin⁡2t+cos⁡2t+(cos⁡2t−sin⁡2t)2|\mathbf{r}'(t)| = \sqrt{\sin^2 t + \cos^2 t + (\cos^2 t - \sin^2 t)^2}. What should they check before proceeding?
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