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You are not logged in. #1 20130304 06:19:26
Gabriel's HornAccording to the textbook, the surface area of the curve y=1/x for x>=1, rotated around xaxis is infinite. Here we have a=1, b=\infty, and f(x)=1/x Since one of the bounds is infinity, we have an improper integral and have to do it with a limit: Looking at the description of Gabriel's Horn in Wikipedia, I see that they used for the surface a function: Why is that? How did they manage to convert into 1? Last edited by White_Owl (20130304 06:22:07) #2 20130304 09:56:49
Re: Gabriel's HornWhere did you get 4u^6 in the denominator in the step right after the substitution from? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #4 20130304 10:44:41
Re: Gabriel's HornMaybe a substitution v=sqrt(u)? The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #5 20130304 10:48:00
Re: Gabriel's HornHi; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #6 20130304 11:06:17
Re: Gabriel's HornAnother attempt: Let Then: We have a formula #24 in the table of integrals in the textbook: So: And here we have first limit is infinity divided by infinity, second limit is infinity and a constant. Therefore we have an infinity in the final answer... Did I make any mistakes? #7 20130304 11:12:23
Re: Gabriel's HornI would say that that is divergent, then. The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment #8 20130304 11:19:11
Re: Gabriel's HornIt is definitely divergent. The integral does not exist. In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #10 20130304 17:35:50
Re: Gabriel's HornWell, 1/x *sqrt(1+1/x^4) is everywhere greater than 1/x, so its integral on any interval will be greater than the integral of 1/x on the same interval! The limit operator is just an excuse for doing something you know you can't. “It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman “Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment 