Visualizing Line Integrals

Question

I have a plane curve $C$ described by parametric equations $x(t)$ and $y(t)$ and a function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. The line integral of $f$ along $C$ is the area of the "fence" whose path is governed by $C$ and height is governed by $f$.

How can I generate a picture of the "fence" in Mathematica?

For the sake of a concrete example, let's borrow from Stewart (since I already borrowed his picture). For $0 \leq t \leq \pi$, define $$ \begin{align*} x(t) &= \cos t\\ y(t) &= \sin t\\ f(x,y) &= 2 + x^2y \end{align*} $$ so that $$ \begin{align*} f(x(t),y(t)) &= 2 + \cos^2 t \sin t. \end{align*} $$

Answer 1

 ListPointPlot3D[
                 Table[{Cos[t], Sin[t], 2 + Sin[t] Cos[t]^2} ,{t, 0, π, 0.01}] , 
                 Filling -> 0]

Answer 2

opts = {MeshFunctions -> (#4 &), 
        MeshShading ->   {{Opacity[#2], #1}, {Opacity[#2/2], #1}}, 
        BoxRatios ->     {1, 1, 1/2}, 
        BoundaryStyle -> Directive[Thin, Blue]} &; 
Show[
 ParametricPlot3D[{Cos[t], Sin[t], z (2 + Sin[t] Cos[t]^2)}, {t, 0, π/2}, {z, 0, 1}, 
                  Evaluate@opts[Green, .4]], 
 ParametricPlot3D[{Cos[t], z Sin[t], 0 }, {t, 0, π/2}, {z, 0, 1},
                  Evaluate@opts[Blue, .2]],
 ParametricPlot3D[{z Cos[t], Sin[t], 0 }, {t, 0, π/2}, {z, 0, 1}, 
                  Evaluate@opts[Blue, .2]]]

Answer 3

I have not done the labeling but this is a start:

axes[n_] := 
  With[{uv = n IdentityMatrix[3]}, 
   Graphics3D[{Arrow[{{0, 0, 0}, #}] & /@ uv, 
     MapThread[
      Text[#1, 1.1 #2] &, {Style[#, 20] & /@ {"x", "y", "z"}, uv}]}, 
    Boxed -> False]];
p = ParametricPlot3D[{Cos[t], Sin[t], u (2 + Cos[t]^2 Sin[t])}, {t, 0,
     Pi/2}, {u, 0, 1}, MeshFunctions -> {#4 &, #5 &}, 
   Mesh -> {10, {0.99}}, 
   MeshStyle -> {Directive[Blue, Thick], 
     Directive[Red, Thickness[0.01]]}, Boxed -> False, 
   PlotStyle -> {LightBlue, Opacity[0.5]}, Axes -> False];
Show[axes[3], p]

Answer 4

This is just a small tweak of Belisarius' answer, using MeshFunctions to get the vertical lines and BoundaryStyle to get the "fence".

ParametricPlot3D[{Cos[t], Sin[t], z (2 + Sin[t] Cos[t]^2)}, {t, 
  0, \[Pi]/2}, {z, 0, 1}, MeshFunctions -> {#2 &}, MeshStyle -> {Red},
  BoundaryStyle -> Directive[Thick, Blue], BoxRatios -> {1, 1, 1/2}, 
 PlotStyle -> Opacity[.5]]

An alternative that gives more "even" vertical lines is to specify the mesh in terms of t and z.

ParametricPlot3D[{Cos[t], Sin[t], z (2 + Sin[t] Cos[t]^2)}, {t, 
  0, \[Pi]/2}, {z, 0, 1}, 
 Mesh -> {Range[0, 1, 0.1], Range[0, 2, 0.5], Range[0, 1, 0.1]}, 
 MeshStyle -> {Directive[Red, Thick], Directive[Green, Thick]}, 
 BoundaryStyle -> Directive[Blue, Thick], BoxRatios -> {1, 1, 1/2}, 
 PlotStyle -> Opacity[.5]]

And as Belisarius points out in comments (I'm going to have to CW this one!), MeshFunctions -> {#4 &} is even evener :)

ParametricPlot3D[{Cos[t], Sin[t], z (2 + Sin[t] Cos[t]^2)}, {t, 
  0, \[Pi]/2}, {z, 0, 1}, MeshFunctions -> {#4 &}, MeshStyle -> {Red},
  BoundaryStyle -> Directive[Thick, Blue], BoxRatios -> {1, 1, 1/2}, 
 PlotStyle -> Opacity[.5]]