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 $$ \begin{align*} f(x(t),y(t)) &= 2 + \cos^2 t \sin t. \end{align*} $$
ParametricPlot3D
. $\endgroup$ – user484 Nov 10 '14 at 4:18