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Be $\gamma: \mathbb{R}\rightarrow \mathbb{R}^3$ defined by:

$$\gamma(\theta):= (\cos(\theta), \sin(2\theta), \cos(3\theta))$$

and be $$S:=\lbrace t\gamma (\theta): t\in [0,1], \hspace{0.1cm} \theta\in\mathbb{R}\rbrace$$

Calculate the area of $S$.

I have problems understanding this exercise... When I have a parametric curve defined by $x=f(t)$ and $y=g(t)$, $\alpha \le t\le \beta$ I use the formula: $$\int_{\alpha}^{\beta} f'(t)g(t)dt$$ (I understand the formula and his demonstration). But I don't know if exists a generalization to calculate the area of any parametric curve, or if this exercise needs a different kind of approach.

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  • $\begingroup$ I believe that you want $x = f(t)$, $y= g(t)$? $\alpha(t)$ doesn't make sense to me. $\endgroup$ – Calvin Lin Jun 3 '13 at 4:58
  • $\begingroup$ Oh, sorry, I'll fix now, was a mistake in notation. $\endgroup$ – El Peluca Sabe Jun 3 '13 at 5:06
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Hint: the area element for a parametric surface ${\bf R} = {\bf R}(u,v)$ is $\left| \dfrac{\partial {\bf R}}{\partial u} \times \dfrac{\partial \bf R}{\partial v}\right|\; du\; dv$

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  • $\begingroup$ Following your hint, I need to calculate $\left|(cos(\theta), sin(2\theta), cos(3\theta))\times (-tsin(\theta), 2tcos(2\theta), -3tsin(3\theta))\right|d\theta dt$? $\endgroup$ – El Peluca Sabe Jun 3 '13 at 5:27

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