%M148 W2015 A3
\documentclass[12pt]{amsart}
\usepackage{amssymb}
\usepackage{graphicx}
%\usepackage{fullpage}
\topmargin -0.5in
\textheight 9in
\textwidth 6.5in
\oddsidemargin .0in
\evensidemargin .0in

%      Blackboard bold letters
\newcommand{\bC}{{\mathbb{C}}}
\newcommand{\bN}{{\mathbb{N}}}
\newcommand{\bQ}{{\mathbb{Q}}}
\newcommand{\bR}{{\mathbb{R}}}
\newcommand{\bZ}{{\mathbb{Z}}}
%Greek Letters
\newcommand{\de}{\delta}
\newcommand{\ep}{\varepsilon}
\renewcommand{\phi}{\varphi}
%
\renewcommand{\P}{{\mathcal{P}}}
\newcommand{\Q}{{\mathcal{Q}}}

%      Text used in equations
\newcommand{\AND}{\text{ and }}
\newcommand{\FOR}{\text{ for }}
\newcommand{\FORAL}{\text{ for all }}
\newcommand{\IF}{\text{ if }}
\newcommand{\IFF}{\text{ if and only if }}
\newcommand{\OR}{\text{ or }}
\newcommand{\qand}{\quad\text{and}\quad}
\newcommand{\qor}{\quad\text{or}\quad}
\newcommand{\qfor}{\quad\text{for}\quad}
\newcommand{\qforal}{\quad\text{for all}\quad}
\newcommand{\qif}{\quad\text{if}\quad}
\newcommand{\qiff}{\quad\text{if and only if}\quad}

\newcommand{\dlim}{\displaystyle\lim\limits}
\newcommand{\dint}{\displaystyle\int}
\newcommand{\dsum}{\displaystyle\sum\limits}
\newcommand{\upminus}{\raise.9ex\hbox{-\!}}

\newcounter{exerci}
\renewcommand{\theexerci}{\arabic{exerci}.}
\newcounter{exercii}
\renewcommand{\theexercii}{(\arabic{exerci}) }
\newenvironment{Ex}{\begin{list}%
 {\theexerci\hfill}{\usecounter{exerci}\rightmargin=0pt\leftmargin=12pt%
 \labelwidth=12pt\labelsep=6pt\itemsep=3ex}}{\end{list}\medbreak}
\newenvironment{parts}{\vspace{1ex}\begin{enumerate}\renewcommand{\itemsep}{2ex}
 \renewcommand{\labelenumi}{(\alph{enumi})}}{\end{enumerate}}
\newcommand{\hint}{\textbf{Hint:} }

\begin{document}
\thispagestyle{empty}

%%%%%%   TITLE   %%%%%%
\begin{center}
\textbf{\large Math 148 Assignment 3}\\[1ex]
\textbf{Due  2:00 p.m.\ Monday, February 9 in the Math 148 dropbox.}  
\end{center}

\vskip 2em 
\begin{Ex}

\item Evaluate the following improper integrals when they exist.
\begin{parts}
\item $\dint_2^\infty \dfrac{dx}{x(\log x)^a}$ for $a>0$. 

\item $\dint_0^{\pi/2} \log \sin x \,dx$. \hint substitute $u= \frac\pi2 - x$ and combine.
\end{parts} 

\item  Which of the following improper integrals exist? (Do not try to evaluate them exactly.) \\[2ex]
(a) $\dint_0^\infty \dfrac1{\sqrt x} \,\sin \tfrac1x \,dx$ \qquad\qquad
(b) $\dint_0^1 \dfrac{dx}{\ln x}$ \qquad\qquad 
(c) $\dint_\pi^\infty \dfrac{\sin x}{\log x} \,dx$ \\


\item Suppose that $f(x)$ and $g(x)$ are bounded continuous functions on $[0,\infty)$.
If $\int_0^\infty f(x) \,dx$ exists as an improper integral, 
does it follow that $\int_0^\infty f(x)g(x) \,dx$ also exists?
Give a proof or provide a counterexample.


\item  
\begin{parts}
\item Consider the region $R$ bounded by the curve $y=\dfrac1{\sqrt{7x-10-x^2}}$ for $2 < x < 5$
together with the lines  $x=2$,  $x=5$ and $y=0$.
Compute the volume of the solid obtained by rotating $R$ about the $y$-axis.
\item Consider the region $S$ bounded by the curve $y=\log x$ for $0 < x \le 1$
together with the lines  $x=0$ and $y=0$.
Compute the volume of the solid obtained by rotating $S$ about the $x$-axis.
\end{parts}


\item Consider the parabola $P$ given by $y = ax^2$ for $a>0$. 
At each point $(x_0,y_0)$ on $P$, construct the \textit{normal} line through $(x_0,y_0)$ 
perpendicular to the tangent line, and consider the area of the sector of $P$ cut off by this line.
\begin{parts}
\item Find the minimal area of this sector.
\item What are the slopes of the normal lines that minimize this area?
\end{parts} 


\item \begin{parts}
\item Compute the arc length of the curve $y=x^2$ from $x=0$ to
$x=1$.

\item Show that the arc length of the curve $y=x^p$ 
from $x=0$ to $x=1$ is an increasing function of $p$ for $p \ge 1$.\\
{\bf Warning:} as far as I know, this cannot be done using the arc length formula.\\
A geometric argument is needed.
\end{parts} 

\end{Ex}
\end{document}
