Arc length of z=Ay^2-By^3

Jans123

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Mar 28, 2017
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Hi guys,

I'm getting stuck in trying to come up with a formula for the arc length of my equation, z = A*y^2 - B*y^3. A and B are both constants. I apologise for using z and y, this is just the way I have my problem set up (z is my vertical axis, and y is my horizontal axis: I guess you can treat it as y = A*x^2 - B&x^3).

Ideally, I would like to compute the arc length formula for this equation, set it to a certain number, e.g. 10 and calculate the corresponding horizontal value (on my y-axis).

Thank you for your help!

Jan :D
 
I would presume that you know that, given z= f(y), the arc-length is given by \(\displaystyle \int \sqrt{\left(\frac{dz}{dy}\right)^2+ 1} dx\). What does that give you?
 
Thanks for replying so quickly! I'm currently doing:
1.
\(\displaystyle z = Ay^2 - By^3 \)
2.
\(\displaystyle \frac{dz}{dy} = 2Ay - 3By^2 \)
3.
\(\displaystyle Arc Length = \int \sqrt{(2Ay - 3By^2)^2+ 1} dy\)
4.
\(\displaystyle Arc Length\) from \(\displaystyle 0\) to \(\displaystyle y=10\) equals \(\displaystyle \int_0^{10} \sqrt{(2Ay - 3By^2)^2+ 1} dy\)

Unfortunately, this doesn't actually compute (i'm using Maple 2015). The calculation runs and runs and runs - I was wondering whether to convert stage 3 from the \(\displaystyle \sqrt{x^2+1}\) (where \(\displaystyle x=\) (2.)) format to an approximation using a series, but this seems to be very temperamental and fails.

Any ideas?

Thank you.
 
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