In this laboratory we will explore the concept of arc-length.
We begin with the example of the semi-circle.

F1[x_]:= Sqrt[1-x^2]
F1[x]
F1p[x_] := -x/Sqrt[1-x^2]
F1p[x]

We may plot a function with the following command. Note the set backets that pair functions and also pair the limits.

Plot[{F1[x]},{x,-1,1}, AspectRatio -> Automatic]

We know how to compute the integral for the power rule for instance

A1:=Integrate[Sqrt[1+F1p[x]^2],{x,-1,1}]
A1

A2:=NIntegrate[Sqrt[1+F1p[x]^2],{x,-1.0,1.0}]
A2

A3:=NIntegrate[Sqrt[1+F1p[x]^2],{x,-1.0,1.0},PrecisionGoal->12]
A3

Now we compute the result numerically ourselves for n=10, 100, 1000, 10000, 100000

n:=10
a:=-1.0
b:=1.0
h:= (b-a)/n
A2 :=NIntegrate[Sqrt[1+F1p[x]^2],{x,a,b}]
RS:=Sum[Sqrt[1+ F1p[a+(i-0.5)*(b-a)/n]^2]*(b-a)/n,{i,1,n}]
PE:=Abs[(RS-A2)/A2]*100
Print["The Integral is ",A2," Riemann sum is ", RS," the Percent error is ", PE, " for n = ", n]

Fill in the chart
n....................Int...................RS.................PE
10                   Pi                  2.76             12.115
100
1000
10000
100000

When finished  use numerical methods  and the code Below to find the arc-length 
for F[x_] := Sqrt[1-x^4] on  [a,b] given as

G1[x_] := Sqrt[1-x^4]
G1p[x_] := 1
n:=10
a:=-0.9999
b:=0.9999
h:= (b-a)/n
B2 :=NIntegrate[Sqrt[1+G1p[x]^2],{x,a,b}]
RS2:=Sum[Sqrt[1+ G1p[a+(i-0.5)*(b-a)/n]^2]*(b-a)/n,{i,1,n}]
RS2
PE2:=Abs[(RS2-B2)/B2]*100
Print["The Integral is ",B2," Riemann sum is ", RS2," the Percent error is ", PE2, " for n = ", n]

When finished  use numerical methods  and the code above to find the arc-length 
for F[x_] := Sqrt[1-x^4] on  [a,b] given as

1) [-0.9999, 0.9999]

n....................Int...................RS.................PE
10                   *                      *                   *
100
1000
10000
100000

2)  [ 0, 0.9999] for F[x_] := Sqrt[1-x^4]


n....................Int...................RS.................PE
10                   *                      *                   *
100
1000
10000
100000


3)[ 0.5 , 0.9999 ] for F[x_] := Sqrt[1-x^4]


n....................Int...................RS.................PE
10                   *                      *                   *
100
1000
10000
100000

when finished email this completed file  to 

prcoulton@eiu.edu