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\title{Calculus and Elements of Linear Algebra I}
\author{Homework 8}
\date{Due on Moodle, Monday, November 9, 2020}
\maketitle


\begin{enumerate}

\item Find
\[
  \int_0^1 \ln x \, \d x \,.
\]

\item Show that
\[
  \int_0^\infty \frac{1}{\sqrt{1+x^8}} \, \d x
\]
is convergent.

\emph{Hint:} There is no elementary way to evaluate this integral.
However, to only \emph{test} convergence, you can bound the integrand
by a simpler function and use the following fact without proof: Let
$f \colon [a, \infty) \to \R$ be a bounded and increasing function.
Then $\lim_{x \to \infty} f(x)$ exists.

\item \begin{enumerate}
\item Show that the volume enclosed when revolving the curve $y=f(x)$,
where $f \colon [a,b] \to [0,\infty)$ about the $x$-axis in
three-dimensional $x$-$y$-$z$ space is given by
\[
  V = \pi \int_a^b f^2(x) \, \d x \,.
\]

\item Compute the volume of the solid obtained by revolving the graph of
of $y=x^3$ on $[0,1]$ about the $x$-axis.
\end{enumerate}

\item
\begin{enumerate}
\item Compute the volume of the infinite solid obtained by revolving
the graph of $y=x^{-2/3}$ on $[1,\infty)$ about the $x$-axis and show
that it is finite.


\item Show that the cross-sectional area when cutting this solid in
the $x$-$y$ plane is infinite.
\end{enumerate}

\item Hook's law states that the force exerted by an ideal spring when
extended from its equilibrium position at $x=0$ to length $x$ is given
by
\[
  F(x) = - k \, x \,,
\]
where $k$ is a positive constant characterizing the stiffness of the
spring.  Compute the work required to expand the spring from its
equilibrium position to length $\ell$.

\end{enumerate}

\end{document}
