# How can I plot the integral of a unit tangent vector? [closed]

The questions states:

Let

r[t_]:= {E^(-t), 3t^2, 4 Sin[t]}


Plot and compare r[t] and the integral of T[t], T[t] being the unit tangent vector.

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## closed as off-topic by Daniel Lichtblau, belisarius, Artes, Kuba, m_goldbergSep 19 '13 at 21:12

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Which question states that? Is it HW? –  belisarius Sep 19 '13 at 20:18
This question appears to be off-topic because it is about math homework, not to do with the software Mathematica. –  Daniel Lichtblau Sep 19 '13 at 20:23

Can't have a question like this without Manipulate:

r[t_] := {Exp[-t], 3 t^2, 4 Sin[t]}
unitT[t_] := Normalize[r'[t]];
(* quicker to solve a DE once than calculate integral over and over *)
int = Module[{y, t},
y /. First@NDSolve[{y[-Pi] == r[-Pi], y'[t] == unitT[t]}, y, {t, -Pi, Pi}]]

original = ParametricPlot3D[
r[t], {t, -Pi, Pi},
AxesLabel -> {x, y, z},
BoxRatios -> 1,
unitized = ParametricPlot3D[
int[t], {t, -Pi, Pi},
AxesLabel -> {x, y, z},
BoxRatios -> 1,

Manipulate[
GraphicsRow[{
Show[{
original,
Graphics3D[Arrow[{r[tt],r[tt] + r'[tt]}]]}]
,
Show[
unitized,
Graphics3D[Arrow[{int[tt],int[tt] + unitT[tt]}]]
]}],
{tt, -Pi, Pi}]


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I'm not really sure what you want, but since you should figure that out yourself anyway here is the most direct translation I can figure:

r[t_] := {E^(-t), 3 t^2, 4 Sin[t]}

Block[{t},
f[t_] = Integrate[r[t], t]
];

ParametricPlot3D[{r[t], f[t]}, {t, 0, 10}, BoxRatios -> 1]

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