Derivation of Integral Arc Length Formula

[El Moriana]

Homework Statement

My textbook [Engineering Mathematics, Stroud, 6th Edition, page932] runs through the derivation of the integral formula for arc length. I got confused at one of the steps:

[partial](ds/dx)=sqrt(1+([partial](dy/dx))^2)
if [partial]dx tends to 0,
ds/dx=sqrt(1+log(dy/dx)^2)
s=int_a^b(sqrt(1+(dy/dx)^2))

Where does the log come from and where does it go?

Homework Equations

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The Attempt at a Solution

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[rock.freak667]

Could you post a scan of that page or at least that section with the derivation?

I don't think that log is supposed to be there.

How I learned it was in a curve if you join any two points, the 'x' distance would be Δx and the corresponding 'y' distance would be Δy. The chord length would be related as

(ΔS)²= (Δx)² +(Δy)²

or

(ΔS/Δx)² = 1 + (Δy/Δx)²

as Δx→0, Δy/Δx = dy/dx and ΔS/Δx = dS/dx

so

\frac{dS}{dx} = \sqrt{1+ \left( \frac{dy}{dx} \right)}

which you can then integrate.

 

[El Moriana]

Here we go, apologies for the messy formula earlier, I didnt realize there was a Latex button.
I was expecting exactly what you typed in the derivation, hence the confusion on my part.

 

[rock.freak667]

I really don't think that 'log' should be there. Else the next step does not make sense.

 

[El Moriana]

Ok, well thanks for looking it over.