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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Second Practice on Improper Integrals}
\centerline{April 4, 2018}
\vskip 10pt
\noindent
{\it Background}
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Whenever $a = -\infty$ or $b = +\infty$ or both, the integral 
$$\int_a^b f(x)\ dx$$
is also said to be an \emph{improper integral}. As in the cases we saw yesterday,
these new improper integrals are also handled by \emph{taking limits of
``ordinary'' integrals}.  
\begin{itemize}
\item We say $\int_a^\infty f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{b \to \infty} \int_a^b f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item We say $\int_{-\infty}^b f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{a \to -\infty} \int_a^b f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item We say $\int_{-\infty}^\infty f(x)\ dx$ \emph{converges}
if, splitting at any finite value $c$ (often at $c = 0$), \emph{both of the limits} here: 
$$\lim_{a \to -\infty} \int_a^c f(x)\ dx + \lim_{b \to +\infty} \int_c^b f(x)\ dx$$ 
exist and we say the integral \emph{diverges} otherwise.  (Note this says that if either one
of the two limits does not exist, then the integral diverges.)
\end{itemize}
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\noindent
{\it Questions}
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For each of the following,
\begin{enumerate}
\item[(i)] Note which of the limits of integration are infinite.
\item[(ii)] Set up the appropriate limit(s) and evaluate.
\item[(iii)]  Say whether the integral converges or diverges.
\end{enumerate}
\begin{enumerate}
\item[(1)] $\displaystyle \int_{-\infty}^2 e^{10x}\ dx$
\item[(2)] $\displaystyle \int_4^{\infty} \frac{1}{x^2 + 16}\ dx$
\item[(3)] $\displaystyle \int_{-\infty}^\infty \frac{1}{x^2 + 16}\ dx$
\item[(4)] Integrals over infinite intervals can lead to somewhat counterintuitive statements.
For example, explain why 
\begin{figure}
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\includegraphics[width=.5\linewidth]{InfiniteHorn.png}
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\caption{The region for problem 4.}
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\begin{enumerate}
\item[(a)]  The area between the graph $y = \frac{1}{x}$ and the $x$-axis for $x \ge 1$
is not finite, but
\item[(b)]  When the area from part (a) is rotated about the $x$-axis, it forms a 
solid of revolution of finite volume(!)
\end{enumerate}
\end{enumerate} 
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\begin{figure}
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\includegraphics[width=.8\linewidth]{needleregion.png}
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\caption{The region for problem 4.}
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