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\title{MATH3210 - Spring 2022 - Section 004}
\author{Homework 8}

\begin{document}

\maketitle

\begin{problem}
We say that a partition $\mc P$ of $[-a,a]$ is {\it symmetric} if $0 \in \mc P$ and whenever $x \in \mc P$, $-x \in \mc P$. Let $f : [-a,a] \to \R$ be an integrable function. Show that for every $\ve > 0$, there exists a symmetric partition $\mc P$ such that $U(f,\mc P) -\ve < \int_{-a}^a f\, dx < L(f,\mc P) + \ve$.
\end{problem}


\begin{problem}
Let $f : [-a,a] \to \R$ be an {\it even} function. That is, a function such that $f(x) = f(-x)$. Show that if $f$ is integrable, then

\[ \int_{-a}^a f\, dx = 2\int_{0}^a f\, dx. \]

[{\it Hint}: Use the previous problem]
\end{problem}

\begin{problem}
Show directly that the function $f(x) = x^2$ is integrable on $[-1,1]$, and compute $\displaystyle\int_{-1}^1 f\, dx$. {\it Do not appeal to the theorem that every continuous function is integrable, every monotone function is integrable or the fundamental theorem of calculus}. You may use the previous problem and the following formula:

\[ \sum_{k=1}^{n-1} k^2 = \dfrac{n(n-1)(2n-1)}{6} \]
\end{problem}

\begin{problem}
Let $f$ be an function on $[a,b]$ such that $\abs{f(x)} \le B$ for all $x \in \R$. 

\begin{enumerate}[i)]
\item Show that $\abs{f(x)^2-f(y)^2} \le 2B\abs{f(x)-f(y)}$ for all $x,y \in [a,b]$.
%\item Show that if $[x_i,x_{i+1}]$ is an interval determined by a partition $\mc P$, then

%\[ M_i(f^2) - m_i(f^2) \le 2B(M_i(f) - m_i(f)), \]

%where $M_i(*)$ and $m_i(*)$ are the supremum and infemum of the function $*$ on $[x_i,x_{i+1}]$, respectively.
\item Show that for any partition $\mc P$ of $[a,b]$, $U(f^2,\mc P) - L(f^2,\mc P) \le 2B(U(f,\mc P) - L(f,\mc P))$.
\item Show that if $f$ is integrable on $[a,b]$, then so is $f^2$.
\end{enumerate}
\end{problem}


\end{document}


