wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> easy >> Infinite, or finite area?
        
(Message started by: Benoit_Mandelbrot on Jan 16^th, 2004, 8:37am)

Title: Infinite, or finite area?
Post by Benoit_Mandelbrot on Jan 16^th, 2004, 8:37am

We have the function ln(x). Is there a finite area above the curve and below the x-axis from 0 to 1, and what is the arc length? How about revolving it around the axises? Is there finite volume, and how about surface area?

Anyone can now post here.

Title: Re: Infinite, or finite area?
Post by THUDandBLUNDER on Jan 16^th, 2004, 9:15am

See also Gabriel's Horn (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_easy;action=display;num=1050956240;start=8).

Title: Re: Infinite, or finite area?
Post by Benoit_Mandelbrot on Jan 16^th, 2004, 9:34am

That has to do with 1/x though, not ln(x), so...

Title: Re: Infinite, or finite area?
Post by Sameer on Jan 16^th, 2004, 9:44am

I wonder who is going to post the answer so that we can start poking our noses (well i don't consider myself in league of this Uber Puzzlers) but this is too easy for me too :P ... so will wait until the author gives a nod ;D

Title: Re: Infinite, or finite area?
Post by Sameer on Jan 16^th, 2004, 9:45am

wow that was my 100th post yippeee 8)

Title: Re: Infinite, or finite area?
Post by John_Gaughan on Jan 16^th, 2004, 4:11pm

Edit: I fixed some of the math. The gerbil wasn't running full speed that day and I erroneously assumed e^x crossed the y-axis at e instead of 1. /Edit

on 01/16/04 at 08:37:57, Benoit_Mandelbrot wrote:

Area
:: [hide]Figuring this out as a logarithm is a bit tough for me, so I chose to flip it around and find the area of e^x from -[infty] to 0.

A = lim (a -> -[infty]) [int] (from a to 0) of e^x dx

A = lim (a -> -[infty]) (e⁰ - e^a)

A = 1 - 0

The area is 1.[/hide] ::

Arc Length
:: [hide]Rather than figure this one out, as I don't remember the answer off the top of my head, my guess is that the arc length is infinite. Even though the domain has a lower bound, the range is unbounded.[/hide] ::

Volume
:: [hide]This is similar to the area problem, except we need to revolve it around the X axis. Since there are no odd shapes, I choose the disk method. The radius of any disk is e^x. The area is [pi]r². The area is then [pi]e^2x. When you integrate this over the length of the function, you get volume.

V = (1/2) [pi] lim (a -> -[infty]) [int] (from a to 0) of e^2x 2dx

V = lim (a -> -[infty]) ([pi]/2) (e⁰ - e^2a)

V = ([pi]/2) (1 - 0)

V = [pi]/2

Surface Area
:: [hide]Rather than explain, since it is similar to the volume, here it goes:

A = 2[pi] lim (a -> -[infty]) [int] (from a to 0) of e^x dx

A = 2[pi] lim (a -> -[infty]) (e⁰ - e^a)

A = 2[pi] (1 - 0)

A = 2[pi]
[/hide]::

Title: Re: Infinite, or finite area?
Post by Sameer on Jan 16^th, 2004, 4:32pm

agh... u can integrate from a to 0 as lnx and e^x are inverses of each other giving area to be 1 and not 1 + e

Title: Re: Infinite, or finite area?
Post by John_Gaughan on Jan 16^th, 2004, 5:28pm

I didn't remember how to do that and couldn't find it when I looked it up. I know it is possible I just didn't remember the rule.

Title: Re: Infinite, or finite area?
Post by John_Gaughan on Jan 29^th, 2004, 6:07am

on 01/16/04 at 16:32:35, Sameer wrote:

agh... u can integrate from a to 0 as lnx and e^x are inverses of each other giving area to be 1 and not 1 + e

You're right, I fixed the solution. For some reason that day I forgot that logarithms/exponentials cross the axis at 1, not the base. Anyway, I don't remember how to integrate a logarithm at x=0, although I suppose I could do (int) e^y dy from -[infty] to 0, which is the exact same thing in terms of the result. So my answer still stands ;-)