Help for a "Surface and Flux integrals" program

[salvomic]

(11-04-2017 10:21 AM)AlexFekken Wrote:  Hi Salvo,

You must paramtrise with polar coordinates from the start; the way you are doing it, your are not integrating over a circle in the (x, y)-space but a square (your image is missing but that is what I think you are doing)

E.g. parametrise the surface as

\[ (x=u \cos(v),\ y = u \sin(v),\ z= u^2)\]
where
\[(u,v) \in [0,1] \times [0,2 \pi] \]

...

\[ |I| = \sqrt{4 u^4 + u^2} = u \sqrt{1 + 4 u^2} \]

etcetera...

It's true, Alex!
the correct input is, therefor:
sfint(1,[u*COS(v),u*SIN(v), u^2],[0,1],[0,2π]) that gives (1/6)*(√5*5*π-π) = 5.3304135, and simplifying we have the same result with (π/6)*(5^(3/2)-1) as in the book (that's correct!)

This fact induce me to think that we should find a way to help user with the changes of coordinates too...

EDIT: add:
But, if the result now correspond a that of the book, why in the Prime, doing this integral \( \int_{\sigma }1dS=\iint\sqrt{4u^2+4v^2+1}dudv=\int_{0}^{1}\int_{0}^{2\pi }\rho \sqrt{1+4\rho ^2}d \rho d\theta = \frac{\pi }{6} (5^\frac{3}{2}-1) \) we get a different result? Am I wrong with its input? or the book's example is not so clear...

Salvo