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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{First Practice on Improper Integrals}
\centerline{April 3, 2018}
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\noindent
{\it Background}
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When $[a,b]$ is a finite interval, the integral 
$$\int_a^b f(x)\ dx$$
is said to be \emph{improper} whenever $f(x)$ has one or more \emph{discontinuities}
in the interval $[a,b]$.  These can happen at the endpoints $a$ or $b$, or at some
$c$ in $(a,b)$.  In all cases, improper integrals are handled by \emph{taking limits of
``ordinary'' integrals}.  For the purposes of this discussion, let us suppose that
$f(x)$ has only one point of discontinuity in the interval.
\begin{itemize}
\item If $f$ is discontinuous at $a$, then we say $\int_a^b f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{c \to a^+} \int_c^b f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item If $f$ is discontinuous at $b$, then we say $\int_a^b f(x)\ dx$ \emph{converges}
if the limit 
$$\lim_{c \to b^-} \int_a^c f(x)\ dx$$ 
exists and we say the integral \emph{diverges} otherwise.
\item If $f$ is discontinuous at $d$ in $(a,b)$, then we say $\int_a^b f(x)\ dx$ \emph{converges}
if \emph{both of the limits} here: 
$$\lim_{c \to d^-} \int_a^c f(x)\ dx + \lim_{c \to d^+} \int_c^b f(x)\ dx$$ 
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)] Say why the integral is improper by locating any discontinuities of the function.
\item[(ii)] Set up the appropriate limit(s) and evaluate.
\item[(iii)]  Say whether the integral converges or diverges.
\end{enumerate}
\begin{enumerate}
\item $\displaystyle \int_1^2 \frac{1}{\sqrt[3]{x - 1}}\ dx$
\item $\displaystyle \int_0^2 \frac{1}{x^2 - 5x + 6}\ dx$
\item $\displaystyle \int_1^3 \frac{1}{x^2 - 4x + 4}\ dx$
\end{enumerate}
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\caption{The region for problem 4.}
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