3
$\begingroup$

I want to solve the following problem, I want to find

$ \iint_S x \, \mathrm{d}S $ where S is the part of the parabolic cylinder that lies inside of the cylinder $z = x^2/2$, and in the first octant of the cylinder $x^2 + y^2 = 1$

I was obviously thinking about switching to cylindrical coordinates, but I have problems setting up the problem and finding the limits.

Could I get some tips / help ? =)

  • 0
    I'd like to help you, but I don't get the picture. This is a 3D plot of your 2 equations. http://cl.ly/1v3J1i2v1V392u352b2g Can you describe to me again on which surface you want to find your integral?2012-05-16

1 Answers 1

1

The main challenge here is finding a suitable parametrization. Since the boundary of the surface of integration is defined in terms of a cylinder, it makes sense to try cylindrical coordinates. We have:

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

Since $\displaystyle z = \frac{x^2}{2}$, the parabolic cylinder has the following parametrization in cylindrical coordinates:

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

And the ranges are:

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

Here is a plot of this parametric representation:

plot

From there, you have a straightforward surface integral to solve.

  • 0
    @N3buchadnezzar As for $dS$, I think you made a mistake there. Can you edit the question to show how you derived that?2012-05-17