calculate

$\displaystyle

\iint_{M}^{}\vec{F}\vec{dS}

$

where

$\displaystyle

\vec{F}=(e^y,ye^x,x^2y)

$

M is a part of hyperboloid $\displaystyle x^2+y^2$

which is located at 0<=x<=1 and 0<=y<=1 ,and its normal vector points outside

like this :

http://i28.tinypic.com/f9p63r.gif

i am used to solve it like this

$\displaystyle

\iint_{M}^{}\vec{F}\vec{dS}=\iint_{D}\frac{\vec{F} \cdot \vec{N}}{|\vec{N}\cdot\vec{K}|}dxdy

$

$\displaystyle

\vec{N}=(2x,2y,-1)

$

$\displaystyle

\iint_{M}^{}\vec{F}\vec{dS}=\iint_{D}\frac{(e^y,ye ^x,x^2y) \cdot (2x,2y,-1)}{1}dxdy

$

now i convert into polar coordinates

x^2+y^2=r

$\displaystyle

=\int_{0}^{2\pi}\int_{0}^{1}\frac{(e^y,ye^x,x^2y) \cdot (2x,2y,-1)}{1}rdrd\theta

$

how to what are the intervals for r

i just guessed its from 0 to 1

i dont know how to know the upper interval here

except that

is this method ok?