In my problem I have to determine the surface of function (rotated around x axis)

f(x)=e^(-x); x E (0,∞) - so x goes from 0 to ∞.

So, if I follow the standard equation for S (y*sqrt(1+(y')2)), I get the result 7,2.
On the other hand, I can divide function into small cylinders with hight of dx. So the integral now will be much simplier:
2 π r dx, where r is the function e^(-x),
The result of the second method is 2π.

If anyone knows why the results differs, please explain.

Thank you!