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\newcommand{\myCourseNumber}{Math 20E}
\newcommand{\myAssignmentDueDate}{Tuesday, November 21, 2022}
\newcommand{\myAssignmentTitle}{Homework Assignment 6}
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\begin{questions}
    
\bigskip

\question Given any two positive constants $c$ and $d$.  Compute the area enclosed by the ellipse

$\displaystyle \left(\frac{x}{c}\right)^2 + \left(\frac{y}{d}\right)^2 = 1$.

\bigskip

\question Find the area of the region between the $x$-axis and the cycloid parametrized by

$\mathbf{r}(t) = \left(t - \sin(t), \ 1 - \cos(t)\right)$ \ with \ $0 \leq t \leq 2\pi$.

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\question %Based on Rogawski 17.1.23
The curve in $\mathbb{R}^2$ satisfying the equation \ $x^3 + y^3 = 3 x y$ \ is called \emph{the folium of Descartes}.
\begin{parts}
\part By setting $y = t x$, verify that \ $\mathbf{c} : \mathbb{R} \setminus \{-1\} \to \mathbb{R}^2$ \ with \ $\mathbf{c}(t) = \left(\frac{3 t}{1 + t^3}, \, \frac{3 t^2}{1 + t^3}\right)$ \ parametrizes the folium of Descartes. (Note: $\mathbb{R} \setminus \{-1\} = (-\infty, -1) \cup (-1, \infty)$.)

\part Use the parametrization above to determine the area enclosed by the loop of the folium. \newline  (Hint: The resulting integral should be an improper integral with $0 \leq t < \infty$.)
\end{parts}

\bigskip

\question %Rogawski 17.1.11
Use Green's theorem to evaluate the line integral of \ $\displaystyle \mathbf{F} = \left(e^{x+y}, \; e^{x-y}\right)$ \ along the clockwise-oriented curve consisting of the line segments joining the points $(0,0)$, $(2,2)$, $(4,2)$, $(2,0)$, and back to $(0,0)$ (and in that order).  Be sure to note the orientation.

(Hint: You might find the change of variables $u = x - y$, $v = y$ helpful when evaluating the double integral.)

\bigskip

\question Let \ $\displaystyle P(x,y) = \frac{-y}{x^2 + y^2}$ \ and \ $\displaystyle Q(x,y) = \frac{x}{x^2 + y^2}$, \ and let $D$ be the unit disk \ $D = \left\{ (x,y) \mid x^2 + y^2 \leq 1 \right\}$.
\begin{parts}
\part Evaluate the area integral \ $\displaystyle \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dx \, dy$ \ over the unit disk $D$.
\part Evaluate the line integral \ $\displaystyle \int_{\partial D} P \, dx + Q \, dy$ \ around $\partial D$, the unit circle with positive orientation.

\smallskip

\part Briefly explain why Green's theorem failed.
\end{parts}


\end{questions}

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