Surface of revolution between two circles I want to compute the surface of revolution between thoose two circles $(x_1,y_1)$ and $(x_2,y_2)$. I assume that the radius of the circles follow a well known law $x(y)$.
What I don't understand is that in my book, they say that an infinitesimal surface is : $2 \pi r(y) ds$ with $ds=\sqrt{dx^2+dy^2}$.
But I would choose $2 \pi r(y) dy$. Indeed if I want to compute the volume I would write : $ \pi r^2(y) dy$, so why should I take $ds$ for the surface and $dy$ for the volume, I don't understand.
(I know that the fact I take $dy$ for the volume is not a justification to take $dy$ for the surface but what I want to know is...how can I know which differential element is the good one ???).
How to be sure of the good differential element to take ?


$dA =2 \pi r(y) ds$ makes no sense of consistency of used notation.
With $(x,y)$ notation of axes $ ds =\sqrt{dx^2+dy^2}$  and surface area $ dA = 2 \pi x ds  = 2 \pi x(y) ds, $ and
with $(r,z)$ notation of axes  $ ds =\sqrt{dr^2+dz^2}$ and  surface area $dA =  2 \pi r ds =  2 \pi r(z) ds$ 
