Arc Length Formula

Date: 07/28/98 at 15:32:34
From: Sue
Subject: Arc length

Find the arc length of a curve y = ln(X^4) for 3 < X < 8. After 
finding the length in symbolic form, find the decimal approximation.

How do you solve this problem?

Date: 07/30/98 at 15:30:48
From: Doctor Margaret
Subject: Re: Arc length

Hi Sue, 

Thanks for writing to us. To solve any problem involving arc length 
and a differentiable function, you need to use the arc length formula.  
This formula is for any function given by y = f(x) that represents a 
smooth curve on the interval [a, b]:

   L = S (a, b) sqrt(1 + (f'(x))^2)

Here, S is used for the integral sign, and (a, b) represents the limits 
of integration, so we are integrating from a to b.   

Before we can use the formula, notice that it contains a derivative 
f'(x) that must be calculated first. The function is y = ln x^4. The 
formula for the derivative of this function is as follows:

   d/dx [ln u] = u'/u.

In your problem, ln u =  ln x^4, and the derivative of x^4 is 4x^3.  
This is u'. Putting this over u we get 4x^3/x^4, which simplifies to 
4/x. This is the derivative f'(x) that we were looking for. Look at the 
original formula given for arc length. 

The next thing before we can do the actual integral is to simplify the 
formula by calculating 1 + f'(x)^2 :

      f'(x) = 4/x

  (f'(x))^2 = (4/x)^2 = 16/x^2

Thus:

   sqrt(1 + (f'(x)^2)) = sqrt(1 + 16/x^2)

Now we have to remember all the rules about square roots and change 
this into something we can use:

   sqrt(1 + 16/x^2) = sqrt((x^2 + 16)/x^2) = sqrt(x^2 + 16)/x

Now you can finally do the integration of sqrt(x^2 + 16)/x from 3 to 8. 

After you do the integration, you will have an algebraic expression for 
the integral. Next, using the numbers a = 3 and b = 8, you compute real 
value by plugging in the value for b first and then subtracting the 
value of the integral for a. For example, the integral of x dx = x^2, 
and from a = 3 to b = 4, the integral takes on the value of 
4^2 - 3^2 = 16 - 9 = 7.

Let me know if you need more help with this. I think the algebra is 
actually the messiest part. One thing about calculus, it makes you get 
really good at algebra. The only thing you have to remember in 
calculus is the formulas. Good luck. 

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