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Read aloud Readability settings Prev Up Next \(\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}}
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Exercises 7.5 Exercises
1. 2.
Define \(s \colon [0,3] \to \R\) by
\begin{equation*}
s(x)=
\begin{cases}
1 & \text{if } 0 \le x \lt 1,\\
-2 & \text{if } 1 \le x \lt 2,\\
3 & \text{if } 2 \le x \le 3.
\end{cases}
\end{equation*}
Show that \(s\) is Riemann integrable and compute
\begin{equation*}
\int_0^3 s(x)\,dx.
\end{equation*}
3. Prove directly from the definitions that the Dirichlet function
\(1_{\Q}\) on
\([0,1]\) is not Riemann integrable.
4.
Show that the function \(f(x)=\lfloor x \rfloor\) is Riemann integrable on \([0,3]\text{,}\) and compute
\begin{equation*}
\int_0^3 \lfloor x \rfloor\,dx.
\end{equation*}
5.
Let \(f \colon [a,b] \to \R\) be bounded, and suppose that \(f(x)=0\) except at finitely many points of \([a,b]\text{.}\) Prove that \(f \in R[a,b]\) and that
\begin{equation*}
\int_a^b f(x)\,dx=0.
\end{equation*}
6.
Let \(f \colon [a,b] \to \R\) be continuous and suppose that \(f(x) \ge 0\) for every \(x \in [a,b]\text{.}\) Prove that if
\begin{equation*}
\int_a^b f(x)\,dx=0,
\end{equation*}
then \(f(x)=0\) for all \(x \in [a,b]\text{.}\)
7.
Define \(F \colon [-1,1] \to \R\) by
\begin{equation*}
F(x)=\int_0^x |t|\,dt.
\end{equation*}
Find an explicit formula for \(F(x)\) and verify that \(F'(x)=|x|\) for every \(x \in [-1,1]\text{.}\)
8.
Let \(f(x)=\operatorname{sgn}(x)\) on \([-1,1]\text{,}\) and define
\begin{equation*}
F(x)=\int_0^x f(t)\,dt.
\end{equation*}
Compute \(F(x)\) explicitly. Show that \(F\) is continuous on \([-1,1]\text{,}\) and explain why \(F\) is not differentiable at \(0\text{.}\)
9.
Suppose \(F\) is differentiable on \([a,b]\text{,}\) that \(F' \in R[a,b]\text{,}\) and that \(F(a)=F(b)=0\text{.}\) Prove that
\begin{equation*}
\int_a^b xF'(x)\,dx=-\int_a^b F(x)\,dx.
\end{equation*}
10.
\begin{equation*}
\int_0^1 x^n\,dx=\frac{1}{n+1}.
\end{equation*}
11.
\begin{equation*}
\int_0^1 x(1+x^2)^4\,dx.
\end{equation*}
12.
Let \(\varphi \in C^1[a,b]\text{,}\) and let \(f\) be continuous on \(\varphi([a,b])\text{.}\) Prove that if \(\varphi(a)=\varphi(b)\text{,}\) then
\begin{equation*}
\int_a^b f(\varphi(x))\varphi'(x)\,dx=0.
\end{equation*}
13. Let
\(f \colon [a,b] \to \R\) be bounded, and suppose that
\(f\) is continuous except at countably many points of
\([a,b]\text{.}\) Use
Theorem E.1.5 to prove that
\(f\) is Riemann integrable.
14.
Let \(h\) be the Thomae function on \([0,1]\text{.}\) Prove that the function \(x \mapsto xh(x)\) is Riemann integrable and that
\begin{equation*}
\int_0^1 xh(x)\,dx=0.
\end{equation*}
