Parametric Surface Area: (x,y,z) = r(u,v) = ((4u)/(u^2+v^2+4), (4v)/(u^2+v^2+4), ...)

[navenoc13]

Guys, I've inserted an attachment containing the question

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The surface is given by:

. . . . .\(\displaystyle (x,\, y,\, z)\, =\, \overrightarrow{r}(u,\, v)\, =\, \left(\dfrac{4u}{u^2\, +\, v^2\, +\, 4},\, \dfrac{4v}{u^2\, +\, v^2\, +\, 4},\, \dfrac{2\, (u^2\, +\, v^2)}{u^2\, +\, v^2\, +\, 4}\right)\)

...where \(\displaystyle (u,\, v)\, \in\, \mathbb{R}^2.\) Compute the surface area of this surface. From the surface area, guess which surface this is.

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The calculation's a bit complicated, so anyone can suggest which method to use for this question?

Need Reply Soon!
Thanks Mate~

 

[tkhunny]

You should try setting up an integration.
Don't forget your Differential Factor.
You may wish to consider methods at the very end of your undergraduate calculus text.
Looks like a Change of Variables may be beneficial. Don't forget your Jacobian.
Okay let's see your best work.

 

[HallsofIvy]

Given a surface defined by x= f(u, v), y= g(u, v), z= h(u,v), we can define the two tangent vectors to the surface a=<f_u, g_u, h_u> and b= <f_v, g_v, h_v>. Their cross product, a x b, is perpendicular to the surface and its length is the "differential of Surface area: dS= |a x b|.