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{\bf Math 222 in class problems} \hspace{.125in} \textbf{Week: March 29, 2021} \hspace{.5in}  \textbf{Name:} %outlier malcom
\begin{enumerate}

%DO CHANGE OF COORDINATES IN WEEK 7

%\item Multi Warm Up \\
%Reparametrize the below curve by arc length from $(1,1,0)$ in the direction of increasing $t$
%\[
%r(t)= \langle e^t,  e^{-t}, t\sqrt{2}   \rangle.
%\]
%Then compute $\mathbf{T}, \mathbf{N}, \mathbf{B}$, $k(s)$, and $\tau(s)$.

  
\item Let $C$ be positively oriented closed curve given by the rectangle with vertices $(0,0), (3,0), (3,4), (0,4).$ Evaluate
 \[
\int_C ye^xdx + 2e^x dy.
\]
A: $4e^3-4$

\item Let $C$ be the positively oriented closed curve given by the circle $x^2+y^2=4$. Evaluate
 \[
\int_C y^3dx - x^3 dy.
\]
A: $-24 \pi$


\item On HW \#8: Fleshing out Jones 12.D \\
Let $S = \{ (x,y) \ | \ a \leq x \leq b, \ 0\leq y \leq f(x)\}$, where $f$ is a non-negative $C^1$ function on $[a,b]$.  Explain how the formula $A= - \int_{\partial S} y dx$ for the area of $S$ in Folland 5.2 Example 3 (Prof Jo Slide 27) leads to the familiar formula $A=\int_a^b f(x) dx.$ (Your argument should be self-contained, e.g. not require the grader to hunt through Jones' book.)


\item Jones Problem 12-4\\
Find the area enclosed by the curve $x^4+y^4=4xy$ in the first quadrant.

\begin{figure}[h]
\begin{centering}
  \includegraphics[width=2in]{p10_figure.png}
  \caption{$x^4+y^4=4xy$}
  \label{fig:boat1}
\end{centering}
\end{figure}


\item On HW \#8 \\
Use Green's theorem as in Folland 5.2 Example 3 (Prof Jo Slide 27) to calculate the area under one arch of the cycloid described parametrically by $\mathbf{r}(t)=\langle R(t-\sin t), R(1-\cos t) \rangle$.  (Folland: $3\pi R^2$). \\


\end{enumerate}


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