Arc Length of a Parabola

pbarclen.html

Exploration

Show that the arc length, s, of a parabola whose parametric equations are

and y=2f t

is given by where

.

Approach

Directly apply the integral definition of arc length.

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

Compute the indefinite integral first.

Clear[f,t];
I1=Integrate[
      Sqrt[D[f*t^2,t]^2+
           D[2*f*t,t]^2],
      t] //Simplify

Evaluate the indefinite integral at the limits.

Clear[t1,t2];
s1=(I1 /. t->t2) - (I1 /. t->t1) //Simplify

The focal length, f, is positive

Clear[E1];
s2=s1 /. Sqrt[f^2*E1_]->f*Sqrt[E1]

Simplify.

s3=Factor[s2]

s4=f*Map[(-1*#)&,s3[[3]]]