Initialization:

SetDirectory["c:\Backup Folder"];
SetDirectory["tech talk"]

distance[pt1_, pt2_] := 
      Sqrt[(pt1[[1]] - pt2[[1]])^2  +  (pt1[[2]] - 
                pt2[[2]])^2  +  (pt1[[3]]  - 
                pt2[[3]])^2]

totaldistance[points_] := 
      Sum[distance[points[[i]], points[[i - 1]]], {i, 2, 
          Length[points]}]

amblight = GrayLevel[0.3];

lightsrc = {{{-0.25,1,1}, GrayLevel[1]}};

standardviewpoint = {3.780, 1.765, 0.779};
viewpoint = standardviewpoint;

axes[xmax_,ymax_,zmax_]:= Graphics3D[{
                               Thickness[0.003],
                               Line[{{0,0,0},{xmax,0,0}}],
                               Line[{{0,0,0},{0,ymax,0}}],
                               Line[{{0,0,0},{0,0,zmax}}]
                               }]

axes[{x,xmin_,xmax_},
         {y,ymin_,ymax_},
         {z,zmin_,zmax_}]:= Graphics3D[{
                               Thickness[0.003],
                               Line[{{xmin,0,0},{xmax,0,0}}],
                               Line[{{0,ymin,0},{0,ymax,0}}],
                               Line[{{0,0,zmin},{0,0,zmax}}]
                               }]

show3dopts = {Boxed->False, 
              Axes->None, 
              DisplayFunction->$DisplayFunction, 
              ViewPoint -> viewpoint,
              AmbientLight -> amblight,
              LightSources -> lightsrc,
              PlotRange -> All};

xticklen = 0.1; yticklen = 0.2; zticklen = 0.1;

xtick[x_]:= Graphics3D[{Thickness[0.003],
                      Line[{{x,-xticklen,0},{x,xticklen,0}}]
                    }];

ytick[y_]:= Graphics3D[{Thickness[0.003],
                      Line[{{-yticklen,y,0},{yticklen,y,0}}]
                    }];

ztick[z_]:= Graphics3D[{Thickness[0.003],
                      Line[{{0,-zticklen,z},{0,zticklen,z}}]
                    }];

Helix-like curve:

t1 = 0.0;

t2 = 4*Pi;

xmin = -1.2; xmax = 1.2;
ymin = -1.2; ymax = 1.2;
zmin = 0.0; zmax = 32.0;

xticks = Table[xtick[i],{i,Ceiling[xmin],Floor[xmax]}];

yticks = Table[ytick[i],{i,Ceiling[ymin],Floor[ymax]}];

zticks = Table[ztick[i],{i,0,30,5}];

f[t_] := {Cos[t], Sin[t], (2/3) t^(3/2)}

curve = 
        ParametricPlot3D[
          Append[f[t], {RGBColor[0, 0, 1], Thickness[0.005]}], {t, t1, 
            t2}];

Integrate[Sqrt[1 + t], {t, 0, 4 Pi}]

f[4 Pi]

N[Integrate[Sqrt[1 + t], {t, 0, 4  Pi}]]

Make animation for Quicktime:

list3d = {};
\(\(Do[
pts = 
          Table[f[t], {t, t1, 
              t2, \((t2 - t1)\)/n}]; 
gpts\  = \ 
          Graphics3D[{PointSize[0.02], 
              Map[Point, pts]}]; 
lines\  = \ 
          Graphics3D[{RGBColor[1, 0, 0], Thickness[0.004], 
              Line[pts]}]; 
pix = 
          Show[
axes[{x, xmin, xmax}, {y, ymin, ymax}, {z, 
                zmin, zmax}], 
curve, gpts, 
            lines, 
xticks, yticks, 
            zticks, 
show3dopts, 
ImageSize -> {600, 
                375}, 
BoxRatios -> {1, 
                1,  .75}, 
DefaultFont -> {"Courier-Bold", 
                18}, 
PlotLabel -> "number of line segments =" <> ToString[
                    PaddedForm[n, 
                      3]]\  <> "
 approx length ="\ \ PaddedForm[
                  N[totaldistance[pts]], {4, 
                    2}]]; 
list3d = 
          Append[list3d, pix], 
{n, 10, 
          50}
];\)

\)