I'm trying to compute the circumference of a wing section. I have broken up the airfoil circumference into arc pieces and used cubic splines to come up with an equation for each piece.
For example, the arc nearest the leading edge of the wing is the function:
y = 0.0290X^3-0.3334x^2+1.8645x+1.4155
To find the length of this arc, I use the Arc Length Formula:
L = ∫ab √(1+[f'(x)])^2dx
The integration limits are: a = 0.75, b = 5
Alternate notation for the arc length formula gives the derivative as dy/dx:
L = ∫ab √(1+[dy/dx])^2dx
The Attempt at a Solution
I started by getting the derivative of the function:
f'(x) = dy/dx = 3(0.0290X^2)-2(0.3334x)+1.8645
I then substituted u for the derivative term:
u = 3(0.0290X^2)-2(0.3334x)+1.8645
Here's where I get bogged down. How do I get du in terms of dx? I tried differentiating u but that gives negative numbers so something is wrong.
I'm wondering if I should try integrating in parts? Or maybe I should first try to factor that polynomial?
Sorry about the lack of math characters. I tried to use latex but it would not work in my browser.