\documentclass[12pt,fleqn]{article}
\pagestyle{empty}


\usepackage{amssymb}
\usepackage{amsmath}


\newcommand{\adj}{\mathrm{Adj}}
\newcommand{\mbf}{\mathbf}
\newcommand{\R}{\mathbb{R}}
\newcommand{\ra}{\rightarrow}
\newcommand{\proj}{\mathrm{proj}}
\newcommand{\dotprod}{\bullet}

\begin{document}

\begin{center}
{\Large TEST 2

Math 202}
\end{center}

\begin{itemize}
\item  Do your work on separate paper.  Make sure your work is neat!
\item  You can use a calculator and book, but no notes!
\item  Your answers need to be justified by your work!
\end{itemize}



\begin{enumerate}

\item  (12 points)
	\begin{enumerate}
	\item  Consider the function $y=x^2$ on $-2 \leq x \leq 1$.
		We rotate this around the line $x=3$ to get a surface.
		Set up (but do not solve!) an integral that represents the area of this surface.
	\item  Take the function $y=x^2$ on $1 \leq x \leq 4$.
		We rotate this around the $y$-axis.
		Find the surface area of the this surface.
	\end{enumerate}

\item  (12 points) Determine if the following series converge or diverge.
	\begin{enumerate}
	\item  \[ \sum_{n=1}^{\infty} (-1)^{n-1} \left( \frac{1}{2} \right)^{1/n} \]
	\item  \[ \sum_{n=1}^{\infty} \frac{n^2}{e^{n^3}}  \]
	\item  \[ \sum_{n=0}^{\infty} \frac{(-1)^n \sqrt{n}}{n+5} \]
	\item  \[ \sum_{n=0}^{\infty} \frac{2^n}{n!} \]
	\end{enumerate}

\item  (6 points) Consider the power series below
\[
f(x) = \sum_{n=0}^{\infty} \frac{5^n x^n}{n^2}
\]
Find the radius of convergence and the interval of convergence of the series.


\item  (10 points) Let $f(x) = \sqrt{x}$.
	Using any method, showing your work, find the first $5$ terms of the Taylor series centered at $x=1$ for the function $f(x)$.


\item  (20 points) Find the following integrals.  
	You can use the integral tables to check your answers, but you find the integrals without the tables.
	\begin{enumerate}
	\item  \[ \int x \ln x \; dx \]
	\item  \[ \int \frac{1}{(x-2)(x+1)} \; dx \]
	\item  \[ \int x \sqrt{9-x^2} \; dx \]
	\item  \[ \int_0^2 \frac{1}{\sqrt{x}} \; dx \]
	\end{enumerate}










\end{enumerate}

\end{document}