Arc Length of a Parabolic Conic Arc

caarclen.html

Exploration

Using exact integration in Mathematica show that the arc length of a parabolic conic arc with control points , and can be expressed exactly in symbolic form in terms of elementary functions of a and b.

Approach

Create the conic arc. Compute the arc length using the standard formula.  Show that the result is a function of a and b only.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create the conic arc.

Clear[a,b];
ca1=ConicArc2D[{0,0},{a,b},{1,0},1/2];

Find the parametric equations.

Clear[t];
{xt,yt}=ca1[t] //Simplify

Compute the derivatives.

{Dx,Dy}=Map[D[#,t]&,{xt,yt}] //Simplify

Integrate to find the arc length.  The resulting function involves elementary functions of a and b only.

I1=Integrate[Sqrt[Dx^2+Dy^2],t];
arclen1=(I1 /. t->1)-(I1 /. t->0) //FullSimplify