Parametrize a surface and calculating a surface integral

[physmatics]

Homework Statement

Calculate the surface integral I = $$\int\int$$ f dS of the function f(x,y,z) = $$\sqrt{1/2 + y^{2}}$$ over the surface S given by $$x^{2} + 2*y^{2} = 1$$, $$0 \leq z \leq x^{2} + y^{2}$$. (Clue: parametrize the surface.)

Homework Equations

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The Attempt at a Solution

The surface is, as far as I can tell, the elliptic cylinder $$x^{2} + 2*y^{2} = 1$$, from z = 0 to z = 1.
Now, I have trouble parametrizing the surface. Can I just parametrize it as an ellips in $$R^{2}$$? The equation of that ellips would be $$x = \sqrt{1 - 2*y^{2}}$$. Then, how do I parametrize the ellips given the equation? And also, why is 'parametrizing the surface' a clue? I really don't get it...
Sorry for clumsy use of LaTeX, I'm not very familiar with it.

Thank you very much!

 

[LCKurtz]

You try to parameterize a surface in a way to express it in a nice form that hopefully makes the integral easy. This problem suggests cylindrical-like coordinates. Here's what I would try. Write your cylinder like this:

$$x^2 + \frac{y^2}{\left(\frac 1 {\sqrt{2}}\right)^2} = 1$$

This suggests

$$x = \cos(\theta),\ y = \frac 1{\sqrt{2}}\sin(\theta)$$

for the ellipse so you could try this parameterization for your surface:

$$\vec R(\theta,z) =\langle \cos(\theta),\frac 1{\sqrt{2}}\sin(\theta),z\rangle$$

Try that. It might get a little messy but I think you will find it works. Be brave.

 

[HallsofIvy]

Strictly speaking, what LKurtz gives is the parametrization of the intersection of that cylinder with a plane of constant z. For the surface itself you will need to add z= z.

You then have the surface parameterized by $x= cos(\theta)$, $y= (1/\sqrt{2})sin(\theta)$, $z= z$ so the two-dimensional surface is in terms of the two parameters $\theta$ and $z$.

We can then write
$$\vec{R}(\theta, z)= \langle cos(\theta), (1/sqrt{2})sin(\theta), z\rangle$$

The derivatives with respect to the two parameters,
$$\vec{R}(\theta, z)_\theta= \langle -sin(\theta), (1/\sqrt{2})cos(\theta), 0\rangle$$
$$\vec{R}(\theta, z)_z= \langle 0, 0 , 1 \rangle$$
are vectors in the tangent plane at each point and their cross product (the "fundamental vector product" the surface), a vector perpendicular to the tangent plane at each point, with "$d\theta dz$" is the vector differential of surface area and its magnitude is the differential of surface area.

 

[LCKurtz]

Strictly speaking, what LKurtz gives is the parametrization of the intersection of that cylinder with a plane of constant z.

Huh? Maybe your scroll-down bar isn't working??