A surface S is obtained by rotation, around the axis OZ, the arc of parabola $\displaystyle \alpha : z = \frac{x^2}{2}, y = 0$ with $\displaystyle 0 \leq x \leq 1$. Determine the area of the surface obtained

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- Jun 3rd 2009, 04:22 PMApprentice123Surface rotationA surface S is obtained by rotation, around the axis OZ, the arc of parabola $\displaystyle \alpha : z = \frac{x^2}{2}, y = 0$ with $\displaystyle 0 \leq x \leq 1$. Determine the area of the surface obtained
- Jun 4th 2009, 01:11 PMApprentice123How do I calculate surface rotation? Must use the matrix?
- Jun 4th 2009, 02:00 PMJester
- Jun 4th 2009, 02:04 PMApprentice123
- Jun 4th 2009, 02:57 PMJester
- Jun 4th 2009, 05:59 PMSpec
Or you can recognize that the rotation will produce an elliptical paraboloid $\displaystyle z=\frac{x^2+y^2}{2}$ which can be parametrized as $\displaystyle \sigma(u,v)=\left(u,v, \frac{u^2+v^2}{2}\right)$

A switch to polar coordinates will then basically produce the same integral as Danny showed you.

$\displaystyle \iint_D \left| \vec {\sigma'_u}\times\vec {\sigma'_v}\right|dudv=\{\text{Polar coordinates}\}=\int \int_0^{2\pi}\sqrt{r^2+1} \cdot rd\varphi dr=$ $\displaystyle 2\pi\int r\sqrt{r^2+1}dr$ - Jun 5th 2009, 01:07 PMApprentice123
- Jun 5th 2009, 04:56 PMApprentice123