How to Find the Length of a Graph Using Calculus?

[cse63146]

Homework Statement

Determine the length of the following graph:

$$f(x) \ = \ \frac{x^5}{10} + \frac{1}{6x^3}$$

Homework Equations

length of a graph: $$\int \sqrt{1 + f'(x)^2}dx$$

The Attempt at a Solution

so $$f'(x) = \frac{x^4}{2} -\frac{1}{18x^4}$$

$$f'(x)^2 = \frac{x^8}{4} + \frac{1}{18} + \frac{1}{324x^8}$$

Is f'(x)^2 correct?

Did I even need to expand, or is there some trick to this?

 

[Dick]

There is almost always a trick to length of arc problems and the trick is almost always make the expression under the radical into a perfect square. Add 1 to f'^2 and you will see that you can. Except fix f'(x) first. I get x^4/2-1/(2*x^4). Why don't you?

 

[cse63146]

I see what I did wrong. I "brought up" the 6 as well, so I wrote it as 6x^-3

 

[Dick]

Ok, so can you square it and then express the expression under the radical as a perfect square? I'm betting you can.