MapleArcLengthSum.mws

Arc Length

Some Approximations

>    with(plots):

>    Int(sqrt(1+(cos(x))^2),x=0..2*Pi);

Int((1+cos(x)^2)^(1/2),x = 0 .. 2*Pi)

>    int(sqrt(1+(cos(x))^2),x=0..2*Pi);

4*2^(1/2)*EllipticE(1/2*2^(1/2))

>    ArcLength:=evalf(%);

ArcLength := 7.640395576

>    SineCurve:=plot([t,sin(t),t=0..2*Pi],thickness=3):

>    segments:=Array(1..4):

>    for i from 1 to 4 do segments[i]:=plot([(i-1)*2*Pi/4+t*2*Pi/4,sin((i-1)*2*Pi/4)+(sin(2*Pi*i/4)-sin(2*Pi*(i-1)/4))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..4));

[Maple Plot]

>    sum(sqrt((2*Pi/4)^2+(sin(2*Pi*n/4)-sin(2*Pi*(n-1)/4))^2),n=1..4);

2*(Pi^2+4)^(1/2)

>    evalf(%);

7.448383556

>    ArcLength;

7.640395576

>    segments:=Array(1..8):

>    for i from 1 to 8 do segments[i]:=plot([(i-1)*2*Pi/8+t*2*Pi/8,sin((i-1)*2*Pi/8)+(sin(2*Pi*i/8)-sin(2*Pi*(i-1)/8))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..8));

[Maple Plot]

>    sum(sqrt((2*Pi/8)^2+(sin(2*Pi*n/8)-sin(2*Pi*(n-1)/8))^2),n=1..8);

(Pi^2+8)^(1/2)+1/2*(Pi^2+16*(1-1/2*2^(1/2))^2)^(1/2)+1/2*(Pi^2+16*(1/2*2^(1/2)-1)^2)^(1/2)

>    evalf(%);

7.580182618

>    ArcLength;

7.640395576

>    segments:=Array(1..12):

>    for i from 1 to 12 do segments[i]:=plot([(i-1)*2*Pi/12+t*2*Pi/12,sin((i-1)*2*Pi/12)+(sin(2*Pi*i/12)-sin(2*Pi*(i-1)/12))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..12));

[Maple Plot]

>    sum(sqrt((2*Pi/12)^2+(sin(2*Pi*n/12)-sin(2*Pi*(n-1)/12))^2),n=1..12);

2/3*(Pi^2+9)^(1/2)+1/3*(Pi^2+36*(1/2*3^(1/2)-1/2)^2)^(1/2)+1/3*(Pi^2+36*(1-1/2*3^(1/2))^2)^(1/2)+1/3*(Pi^2+36*(1/2*3^(1/2)-1)^2)^(1/2)+1/3*(Pi^2+36*(1/2-1/2*3^(1/2))^2)^(1/2)

>    evalf(%);

7.613215923

>    ArcLength;

7.640395576

>    segments:=Array(1..18):

>    for i from 1 to 18 do segments[i]:=plot([(i-1)*2*Pi/18+t*2*Pi/18,sin((i-1)*2*Pi/18)+(sin(2*Pi*i/18)-sin(2*Pi*(i-1)/18))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..18));

[Maple Plot]

>    sum(sqrt((2*Pi/18)^2+(sin(2*Pi*n/18)-sin(2*Pi*(n-1)/18))^2),n=1..18);

4/9*(Pi^2+81*sin(1/9*Pi)^2)^(1/2)+2/9*(Pi^2+81*(sin(2/9*Pi)-sin(1/9*Pi))^2)^(1/2)+2/9*(Pi^2+81*(1/2*3^(1/2)-sin(2/9*Pi))^2)^(1/2)+2/9*(Pi^2+81*(sin(4/9*Pi)-1/2*3^(1/2))^2)^(1/2)+2/9*(Pi^2)^(1/2)+2/9*(P...
4/9*(Pi^2+81*sin(1/9*Pi)^2)^(1/2)+2/9*(Pi^2+81*(sin(2/9*Pi)-sin(1/9*Pi))^2)^(1/2)+2/9*(Pi^2+81*(1/2*3^(1/2)-sin(2/9*Pi))^2)^(1/2)+2/9*(Pi^2+81*(sin(4/9*Pi)-1/2*3^(1/2))^2)^(1/2)+2/9*(Pi^2)^(1/2)+2/9*(P...

>    evalf(%);

7.628266594

>    ArcLength;

7.640395576

>    segments:=Array(1..24):

>    for i from 1 to 24 do segments[i]:=plot([(i-1)*2*Pi/24+t*2*Pi/24,sin((i-1)*2*Pi/24)+(sin(2*Pi*i/24)-sin(2*Pi*(i-1)/24))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..24));

[Maple Plot]

>    sum(sqrt((2*Pi/24)^2+(sin(2*Pi*n/24)-sin(2*Pi*(n-1)/24))^2),n=1..24);

1/3*(Pi^2+144*sin(1/12*Pi)^2)^(1/2)+1/6*(Pi^2+144*(-sin(1/12*Pi)+1/2)^2)^(1/2)+1/6*(Pi^2+144*(-1/2+1/2*2^(1/2))^2)^(1/2)+1/6*(Pi^2+144*(-1/2*2^(1/2)+1/2*3^(1/2))^2)^(1/2)+1/6*(Pi^2+144*(-1/2*3^(1/2)+si...
1/3*(Pi^2+144*sin(1/12*Pi)^2)^(1/2)+1/6*(Pi^2+144*(-sin(1/12*Pi)+1/2)^2)^(1/2)+1/6*(Pi^2+144*(-1/2+1/2*2^(1/2))^2)^(1/2)+1/6*(Pi^2+144*(-1/2*2^(1/2)+1/2*3^(1/2))^2)^(1/2)+1/6*(Pi^2+144*(-1/2*3^(1/2)+si...
1/3*(Pi^2+144*sin(1/12*Pi)^2)^(1/2)+1/6*(Pi^2+144*(-sin(1/12*Pi)+1/2)^2)^(1/2)+1/6*(Pi^2+144*(-1/2+1/2*2^(1/2))^2)^(1/2)+1/6*(Pi^2+144*(-1/2*2^(1/2)+1/2*3^(1/2))^2)^(1/2)+1/6*(Pi^2+144*(-1/2*3^(1/2)+si...

>    evalf(%);

7.633563974

>    ArcLength;

7.640395576

>    segments:=Array(1..30):

>    for i from 1 to 30 do segments[i]:=plot([(i-1)*2*Pi/30+t*2*Pi/30,sin((i-1)*2*Pi/30)+(sin(2*Pi*i/30)-sin(2*Pi*(i-1)/30))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..30));

[Maple Plot]

>    evalf(sum(sqrt((2*Pi/30)^2+(sin(2*Pi*n/30)-sin(2*Pi*(n-1)/30))^2),n=1..30));

7.636020661

>    ArcLength;

7.640395576

>   

>    segments:=Array(1..48):

>    for i from 1 to 48 do segments[i]:=plot([(i-1)*2*Pi/48+t*2*Pi/48,sin((i-1)*2*Pi/48)+(sin(2*Pi*i/48)-sin(2*Pi*(i-1)/48))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..48));

[Maple Plot]

>    evalf(sum(sqrt((2*Pi/48)^2+(sin(2*Pi*n/48)-sin(2*Pi*(n-1)/48))^2),n=1..48));

7.638685490

>    ArcLength;

7.640395576

>    evalf(sum(sqrt((2*Pi/96)^2+(sin(2*Pi*n/96)-sin(2*Pi*(n-1)/96))^2),n=1..96));

7.639967919

>    evalf(sum(sqrt((2*Pi/192)^2+(sin(2*Pi*n/192)-sin(2*Pi*(n-1)/192))^2),n=1..192));

7.640288656

>    evalf(sum(sqrt((2*Pi/1000)^2+(sin(2*Pi*n/1000)-sin(2*Pi*(n-1)/1000))^2),n=1..1000));

7.640391637

>    evalf(sum(sqrt((2*Pi/10000)^2+(sin(2*Pi*n/10000)-sin(2*Pi*(n-1)/10000))^2),n=1..10000));

7.640395515

>    ArcLength;

7.640395576

>    segments:=Array(1..6):

>    for i from 1 to 6 do segments[i]:=plot([(i-1)*2*Pi/6+t*2*Pi/6,sin((i-1)*2*Pi/6)+(sin(2*Pi*i/6)-sin(2*Pi*(i-1)/6))*t,t=0..1],thickness=3,color=blue):end do:

>    display(SineCurve,seq(segments[k],k=1..6));

[Maple Plot]

>    evalf(sum(sqrt((2*Pi/6)^2+(sin(2*Pi*n/6)-sin(2*Pi*(n-1)/6))^2),n=1..6));

7.530014972

>    ArcLength;

7.640395576

>