



 
Gary, That would be "converges when integrated" but your conclusion is correct. However if you integrate 1/x dx which is the formuls on the plane the answer diverges. Obviously not, as you have just proved! Why? The closer an area element is to the axis the less volume it generated upon rotation. The further out you go along the x axis, the closer 1/x gets. So "most of" the infinite area is not able to generate very much volume. Note also that the area of the trumpet is infinite! So it would take an infinite amount of paint to paint the inside, but to fill it completely with paint would only take a finite amount... The resolution of this apparent paradox is that we were assuming a coat of paint of constant thickness, which would soon be too thick to fit into the interior. One of the reasons why this is so counterintuitive is that the integral of 1/x goes to infinity very slowly  we do not have good intuition for logarithmicorder things. Good Hunting! Gary wrote back
Yes, but obviously not without deforming it. RD  


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