Surface integral (stokes?)

I intend to address the adhering to trouble, I intend to locate

$$ \iint_S x \, \mathrm{d}S $$ where S is the component of the allegorical cyndrical tube that exists within the cyndrical tube $z = x^2/2$, and also in the first octant of the cyndrical tube $x^2 + y^2 = 1$

I was clearly thinking of switching over to round works with, yet I have troubles establishing the trouble and also locating the limits.

Could I get some pointers / aid? =)

2022-07-25 20:47:02
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Answers: 1

The major obstacle below is locating an ideal parametrization. Given that the border of the surface area of integration is specified in regards to a cyndrical tube, it makes good sense to attempt round works with. We have:

$$ (x, y, z) = (\rho\cos\phi, \rho\sin\phi, z) $$

Since $\displaystyle z = \frac{x^2}{2}$, the allegorical cyndrical tube has the adhering to parametrization in round coordinates:

$$ (x, y, z) = \left(\rho\cos\phi, \rho\sin\phi, \frac{(\rho\cos\phi)^2}{2}\right) $$

And the arrays are:

$$ \rho \in [0, 1], \phi \in [0, \frac{\pi}{2}] $$

Here is a story of this parametric representation:

From there, you have an uncomplicated surface area indispensable to address.

2022-07-25 22:22:28