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\begin{document}

\centerline{\textbf{Sample Integration Questions}}
\centerline{\textbf{Math 127B}. \textbf{Winter, 2005}}

\bigskip\bigskip
\noindent
\textbf{1.}  Give an example of a function $f : [0,1]\to \Rl$ such that
$f^2$ is Riemann integrable, but $f$ is not.

\bigskip
\noindent
\textbf{2.} Suppose that $f : [a,b] \to \Rl$ is a bounded, Riemann
integrable function. Define $F : [a,b] \to \Rl$ by
\[
F(x) = \int_a^x f(t)\, dt.
\]
Prove that there exists a constant $M$ such that
\[
|F(x)-F(y)|\le M|x-y|\qquad\mbox{for all $x,y\in [a,b]$}.
\]
Is $F$ necessarily differentiable in $(a,b)$?



\bigskip
\noindent
\textbf{3.} Suppose that $g : \Rl \to \Rl$ is continuous.
Define $f : \Rl \to \Rl$ by
\[
f(x) = \int_0^x (x-t) g(t)\, dt.
\]
Prove that $f$ satisfies the following equations:
\[
f^{\prime\prime}(x) = g(x),\qquad
f(0) = f^\prime(0) = 0.
\]

\bigskip
\noindent
\textbf{4.} Define the improper integral
\[
\int_0^\infty \frac{\sin x}{x} \, dx
\]
as a limit of proper integrals, and prove that it converges.

\smallskip\noindent
\textsc{Hint.} Use integration by parts to show that the proper integrals form
a Cauchy sequence.


\bigskip
\noindent
\textbf{5.}
Suppose that
\[
F(x) = \cases{x^2 & for $0\le x < 2$,\cr
x^3 & for $2\le x \le 3$.}
\]
Evaluate the Riemann-Stieltjes integral
\[
\int_0^3 x dF(x),
\]
briefly justifying your computations.

\end{document}