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Topic review (newest first)
Well, 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!
Divergence should be proven or shown...
It is definitely divergent. The integral does not exist.
I would say that that is divergent, then.
We have a formula #24 in the table of integrals in the textbook:
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?
Maybe a substitution v=sqrt(u)?
Now I wonder myself where did I got u^6 Thanks.
Now I am not sure what to do next?
Where did you get 4u^6 in the denominator in the step right after the substitution from?
According to the textbook, the surface area of the curve y=1/x for x>=1, rotated around x-axis 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?