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Length of a Cubic Curve

Date: 12/10/97 at 04:44:06
From: Steve Gardiner
Subject: Length of a cubic curve

Hi Dr. Math,

I have a problem where I need to calculate the TRUE length of a cubic 
curve. I understand that I can use a parametric solution to work out 
many short distances and add them up along the length of the curve, 
but surely this will produce different results dependent upon how 
many instances of t I measure?

Can it be done easily? Or must I use an approximation?

Many thanks, great site by the way...

Date: 12/10/97 at 13:37:00
From: Doctor Rob
Subject: Re: Length of a cubic curve

The answer to your question depends on how the equation is written.
Can you solve for y explicitly? Do you have the curve in parametric
form, such as x = f(t), y = g(t), for some functions f and g of a
parameter t?

If you can solve for y explicitly in terms of x, then the formula for
arc length is

   s = Integral Sqrt[1 + (dy/dx)^2] dx,

where the limits of integration are values of x at the endpoints.

If you cannot solve for y explicitly, you may be able to implicitly
differentiate, and solve for dy/dx explicitly in terms of x alone.  
If so, you can use the above formula.

If you have the parametric situation, then the formula is

   s = Integral Sqrt[(dx/dt)^2 + (dy/dt)^2] dt,
     = Integral Sqrt[f'(t)^2 + g'(t)^2] dt,

where the limits of integration are values of t at the endpoints.

Example:  y = x^3/3 from x = 0 to x = 3.  dy/dx = x^2.

   s = Integral    Sqrt[1 + x^4] dx.

If you can integrate this, you can find the arc length. Oftentimes,
however, the function you end up with is not integrable in closed
form, leading to an elliptic integral or some such. Then all you can
do is evaluate the integral numerically.

-Doctor Rob,  The Math Forum
 Check out our web site!   
Associated Topics:
College Calculus
High School Calculus

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