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\begin{document}
	
	\mbox{}
	\title{Problem Set 8: Arc Length}
	
	\maketitle	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
		
	\begin{problem}
		Find the length of the following curve in the specified intervals.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			
			\item $y = 1 + 6 x^{3/2}, \quad 0 \le x \le 1$;
			
			\item $x = \frac{y^4}{8} + \frac{1}{4y^2}, \quad 1 \le y \le 2$;
			
			\item $x = \frac{1}{3}\sqrt{y}(y-3), \quad 1 \le y \le 9$;
			
			\item $y = \ln(\cos x), \quad 0 \le x \le \pi/3$;
			
			\item $y = \frac{x^2}{4} - \frac{1}{2} \ln x, \quad 1 \le x \le 2$;
			
			\item $y = \sqrt{x - x^2} + \sin^{-1}\sqrt{x}, \quad 0 \le x \le 1/2$;
			
			\item $y = \ln(1 - x^2), \quad 0 \le x \le 1/2$.
			
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		Find the arc length function for the curve $y = \sin^{-1} x + \sqrt{1 - x^2}$ with starting point $(0,1)$.
	\end{problem}
	
	\begin{problem}
		For the function $f(x) = \frac14e^x - e^{-x}$, show that the arc length on any interval has the same value as the area under the curve.
	\end{problem}
	
	\vspace{20pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}