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\title{Math 1a.  Lecture 7 \\
Improper Integrals}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to evaluate integrals of the form $\int_a^\infty f(x) \, dx$, $\int_{-\infty}^a f(x) \, dx$, and $\int_{-\infty}^\infty f(x) \, dx$ (Improper Integrals of Type 1).

\item
To understand and be able to evaluate integrals of the form $\int_a^b f(x) \, dx$, where $f$ is discontinuous at $a$ or $b$ (Improper Integrals of Type 2).

\item
To understand and be able to apply the comparison test for improper integrals.

\end{itemize} 



\section{Improper Integrals of Type 1}

\begin{itemize}

\item 
Evaluate $\displaystyle \int_1^t e^{-x} \, dx$.  Does this integral make sense for all $t$?  What if $t \to \infty$?

\item
Evaluate $\displaystyle \int_1^t \frac{1}{x} \, dx$.  Does this integral make sense for all $t$?  What if $t \to \infty$?

\end{itemize}

Integrals of Type 1 converge to the following limits.  If the limit does not exist, then the integral diverges.
\begin{itemize}

\item
$\displaystyle \int_a^\infty f(x) \, dx = \lim_{t \to \infty} \int_a^t f(x) \, dx$

\item
$\displaystyle \int^a_{-\infty} f(x) \, dx = \lim_{t \to \infty} \int^a_t f(x) \, dx$

\item
$\displaystyle \int_{\infty}^\infty f(x) \, dx = \int^a_{-\infty} f(x) \, dx + \int_a^\infty f(x) \, dx$, provided both of the integrals on the right-hand side converge.  We may use any value for $a$.


\end{itemize}

\section{Improper Integrals of Type 2}

Integrals of Type 1 converge to the following limits.  If the limit does not exist, then the integral diverges.
\begin{itemize}

\item
If $f$ is continuous on $[a, b)$ and discontinuous at $b$, then
$$
\int_a^b f(x) \, dx = \lim_{t \to b} \int_a^t f(x) \, dx.
$$

\item
If $f$ is continuous on $(a, b]$ and discontinuous at $a$, then
$$
\int_a^b f(x) \, dx = \lim_{t \to a} \int_t^b f(x) \, dx.
$$



\end{itemize}


\section{The Comparison Test for Improper Integrals}


Suppose that $f$ and $g$ are continuous functions with $f(x) \geq g(x) \geq 0$ for $x \geq a$.  Then
\begin{itemize}

\item
If $\displaystyle \int_a^\infty f(x) \, dx$ is convergent, then $\displaystyle \int_a^\infty g(x) \, dx$ is convergent.

\item
If $\displaystyle \int_a^\infty g(x) \, dx$ is divergent, then $\displaystyle \int_a^\infty f(x) \, dx$ is divergent.

\end{itemize}

\section{Worksheet Problems}


\begin{enumerate}

\item 
Evaluate $\displaystyle \int_1^t e^{-x} \, dx$.  Does this integral make sense for all $t$?  What if $t \to \infty$?



\item
Evaluate $\displaystyle \int_1^t \frac{1}{x} \, dx$.  Does this integral make sense for all $t$?  What if $t \to \infty$?



\item
Does $\displaystyle \int_1^\infty \frac{1}{x^2} \, dx$ converge?  If so, to what?



\item
Find all values of $p$ such that the integral $\displaystyle \int_1^\infty \frac{1}{x^p} \, dx$ converges.





\item
Does $\displaystyle \int_{-\infty}^\infty \frac{1}{1 + x^2} \, dx$ converge?  If so, to what?





\item
Does $\displaystyle \int_0^1 \frac{x}{\sqrt{1 - x^2}} \, dx$ converge?  If so, to what?






\item
Does $\displaystyle \int_1^\infty \frac{1}{x^5 + 1} \, dx$ converge?  Why or why not?




\item
Does $\displaystyle \int_0^\infty \frac{1}{e^x + x} \, dx$ converge?  Why or why not?



\item
Does $\displaystyle \int_1^\infty \frac{\sin x}{x^2} \, dx$ converge?  Why or why not?



\item
Find all values of $p$ such that the integral $\displaystyle \int_1^e \frac{1}{x (\ln )^p} \, dx$ converges.




\end{enumerate}


\section*{References}

\begin{itemize}

\item
\S 5.10 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{February 22, 2006}



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