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Area & surface integral
https://www.mathimatikoi.org/forum/viewtopic.php?f=23&t=1327
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Author:  Grigorios Kostakos [ Fri Aug 31, 2018 5:53 pm ]
Post subject:  Area & surface integral

Let $E$ be the surface with parametric representation
\begin{align*}
\overline{R}:(-3,3)&\times[0,2\pi]\longrightarrow{\mathbb{R}}^3\,; \quad
\overline{R}(r,\theta)=\left({\begin{array}{c}
\frac{r}{\sqrt{9-r^2}}\,\cos{\theta}\\
\frac{r}{\sqrt{9-r^2}}\,\sin{\theta}\\
\theta
\end{array}}\right)\,,
\end{align*} and the solid cylinder $K: \big\{(x,y,z)\in{\mathbb{R}}^3\;|\; x^2+y^2\leqslant81,\, 0\leqslant z\leqslant 2\pi \big\}$.
  1. Find the area of the surface $S=E\cap K$.
  2. Let the vector field $\overline{F}:{\mathbb{R}}^3\longrightarrow{\mathbb{R}}^3\,;\quad\overline{F}(x,y,z)=\left({x+y+z\,,\,xyz\,,\,y^2}\right)\,.$ Find the surface integral $\oiint_{S}\big(\nabla\times\overline{F}\,\big)\cdot d\overline{S}$.

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