Solving for Arc Length

Date: 9/10/96 at 16:9:0
From: Chris Garver
Subject: Length of Curve

Greetings,

Would you please show me how to solve for the arc length of 
f(x) = 1/3 (x^2 + 2)^(3/2), from [0,2]? 

The answer is supposed to be 14/3, but I'm having difficulty getting 
that result.  I would very much like to see a solution for this 
problem.

Thank you,
  Chris

Date: 9/24/96 at 22:5:27
From: Doctor Jerry
Subject: Re: Length of Curve

Dear Chris,

The arc length formula is the integral from a to b of sqrt(1+y'^2)dx.  
I will not go into a proof of why this holds true (if you have 
questions, feel free to write).

  y'=(1/2)[(x^2+2)^(1/2)]*2x=x*(x^2+2)^(1/2). 

It then follows that

  sqrt(1+y'^2) = sqrt(1+(x^2)*(x^2+2))=sqrt(x^4+2x^2+1).

I feel confident that you can recognize the stuff inside the square 
root as a perfect square. After that, the problem is easy.  I 
get 14/3. 

-Doctor Jerry,  The Math Forum
 Check out our web site!  
http://mathforum.org/dr.math/   

Date: 9/25/96 at 7:48:13
From: Chris Garver
Subject: Re: Length of Curve

Thanks for the help.
  - Chris