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\title{\vspace*{-1.5in}\textbf{Calculus of Polar Curves (Appendix H2)}\\[3pt]\normalsize Thanks to Faan Tone Liu}
\date{}
\author{}

\begin{document}

\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\chead{April 30, 2018}
\rhead{Calculus of Polar Curves}
\vspace{-.7in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item Area of a sector:\\
\begin{minipage}[t]{.3\textwidth}
\vspace{0pt}
\begin{tikzpicture}
\draw(-1.5,0)--(1.5,0);
\draw(0,-1.5)--(0,1.5);
\draw (0,0) circle (1);
\draw (.3,0) arc (0:60:.3);
\draw (0,0)--(0.5,.866);
\draw (.45,.25) node{$\theta$};
\draw (.5,0) node[below]{$r$};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[t]{.5\textwidth}
\vspace{12pt}
Area of entire circle $=$\\[12pt]
Fraction of circle $=$\\[12pt]
Area of sector $=$\\
\end{minipage}

\item Area of a polar region:\\
\begin{minipage}[t]{.35\textwidth}
\vspace{0pt}
\begin{tikzpicture}
\draw(-2,0)--(2,0);
\draw(0,-1.2)--(0,1.2);
\draw(0,0) --(-1.7,.8)node[above]{$\theta_2=b$};
\draw(0,0) --(1.8,.6) node[above]{$\theta_1=a$};
\draw plot[smooth] coordinates {(-1.7,.8) (-1,.9) (0,.7) (0.5,.85)(1.8,.6) };
\end{tikzpicture}
\end{minipage}
\begin{minipage}[t]{.5\textwidth}
\vspace{12pt}
Area of thin slice $=$\\[12pt]
Estimate of area $=$\\[12pt]
Exact area $=$\\
\end{minipage}


\item To find the slopes of tangent lines to polar curves and arc length of polar curves, use parametric equations:
\begin{align*}
x &= \phantom{lasdkjflaskdjflasdkjf}\\\\
y &= \phantom{lasdkjflaskdjflasdkjf}\\\\
\frac{dy}{dx} &= \phantom{the answer is here hereerererer}\\\\
\text{Arc length} &=  \phantom{the answer is here hereerererer} \phantom{the answer is here hereerererer}\\\\
\text{Arc length (simplified)} &=
\end{align*}

\item Other Notes:
\end{itemize}

\newpage
\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item Find the area inside the region bounded by $r = 3 + 3\sin{\theta}$\vfill
\item Find the area of the region that lies inside both $r = 1 + \sin\theta$ and $r = 3\sin\theta$.\vfill
\newpage
\item Find the length of $r = 2\csc\theta$ from $\theta = \frac{\pi}{6}$ to $\theta = \frac{\pi}{2}$. What is the slope of the curve at $x = \frac{\pi}{2}$?
\newpage 
\item Find the arc length of the cardioid $3 + 3\sin\theta$.\vfill
\end{enumerate}
\end{document}
