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\flushleft\underline{Lab 7 --- MATH 1586 Spring 2018} \\
Recall: this site plots surfaces nicely: \href{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/}{http://web.monroecc.edu/manila/webfiles/pseeburger/CalcPlot3D/} \\
Also, it is helpful to use \href{http://www.wolframalpha.com/}{WolframAlpha} to compute integrals. You may also of course use Mathcad to do it. \\

This lab concerns some applications of line integrals. Recall that if $C$ is a curve parametrized by $\vec{r}(t)$ for $a \leq t \leq b$, then
$$\displaystyle\int_C f(x,y,z) \mathrm{d}s = \displaystyle\int_a^b f(x(t),y(t)) \lVert \vec{r}{\hspace{2pt}'}(t) \rVert \mathrm{d}t$$
and if $\vec{F}(x,y,z)$ is a vector field, then
$$\displaystyle\int_C \vec{F} \cdot \mathrm{d}\vec{r} = \displaystyle\int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}{\hspace{2pt}'}(t) \mathrm{d}t.$$
\begin{enumerate}[1.)]
\item Do the following parts.
\begin{enumerate}[a.)]
\item Plot the surface $z=2-x^2-y^2$ and plot where the curve $\vec{r}(t)=\left\langle \cos^3(t), \sin^3(t) \right\rangle$ for $0 \leq t \leq \dfrac{\pi}{2}$ lies on the surface. (i.e. plot $\left\langle \cos^3(t), \sin^3(t), 2-\cos^6(t)-\sin^6(t)\right\rangle$ for $0\leq t \leq \dfrac{\pi}{2}$ using the ``Add to graph: Space curve" drop down menu).
\item Include a picture of your part a in your lab, making sure to also show the curve lying on the surface.
\item Find $\vec{r}{\hspace{2pt}'}(t)$ by using WolframAlpha.
\item Set up the appropriate integral to compute $\displaystyle\int_C 2-x^2-y^2 \mathrm{d}s$ and use WolframAlpha to compute its value.
\end{enumerate}

\item Do the following parts.
\begin{enumerate}[a.)]
\item Plot the vector field $\vec{F}(x,y,z)=\left\langle \sin(y), x \cos(y), xy \right\rangle$ (useful: you'll find the thing for it in the dropdown menu; click checkbox ``use fixed length for all vectors", and mode ``cylindrical array" with ``in each circle" set to 8, ``\# circle" set to 8, and ``along $z$-axis" set to 10). Also plot the space curve $\vec{r}(t)=\left\langle \dfrac{\sin(t)}{(t+1)^2}, \dfrac{\sin(t)}{t+1}, t\sin(3t) \right\rangle$. 
\item Include a picture of what you plotted above.
\item Use WolframAlpha to compute $\vec{r}{\hspace{2pt}'}(t)$. 
\item Write down the integral you would have to solve to compute $\displaystyle\int_C \vec{F} \cdot \mathrm{d}\vec{r}$ and use WolframAlpha to compute its value. 
\end{enumerate}
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