Exploring Analyic Geometry with Mathematica®

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Arc Length of a Parabola

pbarclen.html

Exploration

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

"pbarclen_1.gif" and y=2f t

is given by "pbarclen_2.gif" where

"pbarclen_3.gif".

Approach

Directly apply the integral definition of arc length.

Initialize

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<<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

"pbarclen_4.gif"

Evaluate the indefinite integral at the limits.

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

"pbarclen_5.gif"

The focal length, f, is positive

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

"pbarclen_6.gif"

Simplify.

s3=Factor[s2]

"pbarclen_7.gif"

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

"pbarclen_8.gif"


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