The Riemann Integral

Abstract

The Riemann integral is defined in terms of lower and upper step functions. The major theorems are concerned with characterizations of integrability, the integrability of monotone and continuous functions, the algebra of integrable functions, and the two versions of the fundamental theorem of calculus.

Appendices

Problems

1.1 Problems for Sect. 7.1

• 1. If

  $$ f(x)=\begin{cases} 1 & \text{ if }x>0\\ -1 & \text{ if }x<0 \end{cases}, $$

  then $f$ is integrable on $[-1,1]$ and $\int _{-1}^{1}f=0.$ [Hint: Any lower sum is $\leq 0$ and some lower sum is $=0.$ Hence, $\underline {\int _{-1}^{1}}f=0.]$

• 2. Let $c$ be the Cantor set and let $f:\left [0,1\right ]\to \mathbb {R}$ be determined by

  $$ f(x):=\begin{cases} 1 & \text{when }x\in C\\ 0 & \text{when }x\notin C \end{cases}. $$

  Find an upper step function $S$ for $f$ such that $\sum S<\frac {1}{2}.$

1.2 Problems for Sect. 7.2

• 1. Let $f(x)=2x.$ Use Corollary 7.2.6 to show $f$ is integrable on $[0,2]$ and $\int _{0}^{2}f=4.$

• 2. Let

  $$ f(x)=\begin{cases} 0 & \text{ if }x\notin\{1/n\mid n\in\mathbb{N}\}\\ 1 & \text{ if }x\in\{1/n\mid n\in\mathbb{N}\}. \end{cases} $$

  Prove that $f$ is integrable on $[0,1]$ and $\int _{0}^{1}f=0.$

• 3. Prove a characterization of integrability in the spirit of this section for functions defined on a rectangle.

  The following method for evaluating the integral of $x^{k}$ is due to Pierre de Fermat (17 August 1601 Beaumont-de-Lomagne to 12 January 1665 Castres).

• 4. Fix $k\in \mathbb {N}.$ Let $f(x):=x^{k}$ and $a>1.$ Let $r:=a^{1/n}$. Consider the partition $1<r<r^{2}<\cdots <r^{n-1}<a$ of $[1,a].$

  • a. Write the corresponding upper and lower sums for $\int _{1}^{a}f.$

  • b. Find the limit of these sums as $n\to \infty .$

  • c. Evaluate $\int _{1}^{a}f.$

• 5. In this problem, we assume familiarity with trigonometric functions. Evaluate $\int _{0}^{\pi /2}\sin .$ You may want to prove

  $$ \sum_{k=0}^{n-1}\sin(a+kb)=\frac{\sin(a+(n-1)b/2)\sin)nb/2)}{\sin(b/2)} $$

  and use partitions determined by $x_{i}-x_{i-1}=\frac {\pi }{2n}.$ [Hint: One way to prove the summation formula is to use $e^{it}=\cos (t)+i\sin (t).$]

1.3 Problems for Sect. 7.3

• 1. Suppose $a\leq x_{n}\leq b$ and $(x_{n}-c)$ is null, for some $c\in [a,b].$ Show the characteristic function $\mathbb {1}_{A}$ of $A:=\{x_{n}\mid n\in \mathbb {N}\}$ is integrable on $[a,b]$ and and $\int _{a}^{b}\mathbb {1}_{A}=0.$

• 2. Give an example of a sequence $x_{n}$ such that $0\leq x_{n}\leq 1$ and the characteristic function $\mathbb {1}_{A}$ of $A:=\{x_{n}\mid n\in \mathbb {N}\}$ is not integrable on $[0,1].$

• 3. Let $c$ be the Cantor set. Show the characteristic function $\mathbb {1}_{C}$ of the Cantor set is integrable on $[0,1]$ and $\int _{0}^{1}\mathbb {1}_{C}=0.$ [That is, the “length” of the Cantor set is $0.$]

• 4. Let $f:[0,1]\to \mathbb {R}$ be determined by

  $$ f(x):=\begin{cases} \frac{(-1)^{p}}{q} & \text{when }x=p/q\\ 0 & \text{when }x\not\in\mathbb{Q} \end{cases}. $$

  Show that $f$ is Riemann integrable on $[0,1]$ and $\int _{0}^{1}f=0.$

• 5. Show that a continuous function defined on a rectangle is integrable on that rectangle.

• 6. Consider the points $a_{1}=\frac {1}{2},$ $a_{2}=\frac {1}{4},$ $a_{3}=\frac {3}{4},$ $a_{4}=\frac {1}{8},$ $a_{5}=\frac {3}{8},\ldots .$ Let $f(x)=\sum _{k,a_{k}<x}\frac {1}{2^{k}}$ as in Example 5.1.5. Calculate $\int _{0}^{1}f.$

• 7. Let $f$ and $g$ be continuous real valued functions defined on the compact interval $\left [a,b\right ].$ Suppose $g\geq 0,$ show there is a point $c$ in $\left [a,b\right ]$ such that

  $$ \int_{a}^{b}fg=f(c)\int_{a}^{b}g. $$

1.4 Problems for Sect. 7.4

The purpose of the first three problems below is to show that $f,g$ integrable $\nRightarrow$ $g\circ f$ integrable.

• 1. Let $f:[0,1]\to \mathbb {R}$ be determined by

  $$ f(x):=\begin{cases} \frac{1}{q} & \text{when }x=p/q\\ 0 & \text{when }x\not\in\mathbb{Q} \end{cases}. $$

  Show that $f$ is Riemann integrable on $[0,1]$ and $\int _{0}^{1}f=0.$

• 2. Let $g:[0,1]\to \mathbb {R}$ be determined by

  $$ g(x):=\begin{cases} 1 & \text{when }x>0\\ 0 & \text{when }x=0 \end{cases}. $$

  Show that $g$ is Riemann integrable on $[0,1]$ and $\int _{0}^{1}g=0.$

• 3. Let $f$ and $g$ be as in the preceding two exercises. Note both $f$ and $g$ are functions $[0,1]\to [0,1].$ Let $h:=g\circ f,$ show that $h$ is not integrable on $[0,1].$

• 4. If $f$ is continuous on $[c,d],$ $g$ is integrable on $[a,b],$ and $g([a,b])\subseteq [c,d],$ then $f\circ g$ is integrable on $[a,b].$ [Hint: $f$ is uniformly continuous.]

  It can be show that if $f$ is integrable and $g$ is continuous, then $f\circ g$ need not be integrable. However,

• 5. If $f$ is integrable, $g$ is $\mathcal {C}^{1}$ and $g'(x)\neq 0$ for all $x,$ then $f\circ g$ is integrable. [Hint: $g$ is monotone.]

1.5 Problems for Sect. 7.5

The converse of FTC-Derivative is false, in the sense that differentiability of $g(x)=\int _{a}^{x}f$ does not imply continuity of $f.$

• 1. Give an example of an integrable function $f$ and a point $x_{0}$ such that $g(x)=\int _{a}^{x}f$ is differentiable at $x_{0}$ and $f$ is not continuous at $x_{0}.$

  What can we say about $g$ if $f$ is only integrable? One answer is:

• 2. Suppose $f$ is integrable, prove that $g(x)=\int _{a}^{x}f$ is continuous.

• 3. [Taylor’s formula with integral remainder] If $f$ is $\mathcal {C}^{n+1},$ then

  $$ f(b)=\sum_{j=0}^{n}\frac{f^{(j)}(a)}{j!}(b-a)^{j}+\frac{1}{n!}\int_{a}^{b}(b-x)^{n}f^{(n+1)}(x)\, dx. $$

1.6 Problems for Sect. 7.6

• 1. Suppose $0\leq g(x)\leq f(x)$ for all $x$ and the improper integral $\int _{a}^{\infty }f$ exists. Prove the improper integral $\int _{a}^{\infty }g$ exists.

• 2. Suppose $0<a<1.$

  • a. Show that

    $$ 0\leq\int_{0}^{a}\frac{dx}{\sqrt{1-x^{2}}}\leq\int_{0}^{a}\frac{dx}{\sqrt{1-x}}\leq2. $$

  • b. Show that $I(a):=\int _{0}^{a}\frac {dx}{\sqrt {1-x^{2}}}$ is increasing and bounded by $2.$

  • c. Deduce that $\int _{0}^{1}\frac {dx}{\sqrt {1-x^{2}}}$ exists as an improper integral.

• 3. If $f$ is integrable on $[a,b],$ then

  $$ \int_{a}^{b}f=\lim_{c\nearrow b}\int_{a}^{c}f. $$

  The point of the last problem is that limit used to define the improper integral agrees with the usual integral, when $f$ is integrable.

Solutions and Hints for the Exercises

Exercise 7.1.3. Similar to the proof of Lemma 7.1.1.

Exercise 7.1.7. Any lower step function for $f$ is a lower step function for $g,$ consequently $\underline {\int _{a}^{b}}f\leq \underline {\int _{a}^{b}}g.$

Exercise 7.1.10. If it is not clear how to proceed: examine the proof of Lemma 7.1.11.

Exercise 7.1.12. If $[a,b]\times [c,d]$ is a rectangle, $a=x_{0}<\cdots <x_{m}=b$ is a partition of $[a,b],$ $c<y_{0}\cdots <y_{n}=d$ is a partition of $[c,d],$ and $A_{j,k}$ are real numbers, then $f(x,y):=\sum _{j=1}^{m}\sum _{k=1}^{n}A_{j,k}\mathbb {1}_{]x_{j-1}x_{j}[\times ]y_{k-1}y_{k}[}(x,g)$ is a step function.

Exercise 7.3.4. If $f\neq 0,$ then $f(c)>0$ for some $c.$ By Local Positivity for Limits $f$ is $\geq f(c)/2$ on some open interval $I$ containing $c.$ Let $L$ be the length of $i.$ Then any upper sum for $f$ is $\geq Lf(c)/2.$ Hence, the upper integral of $f$ is $\geq Lf(c)/2.$ Thus, $\int _{a}^{b}f\geq Lf(c)/2.$

Exercise 7.3.5. Equivalent to the Exercise 7.3.4.

Exercise 7.4.2. If $k=0,$ $kf$ is the zero function which has integral zero. The case where $k<0$ is similar to the $0<k$ case, the only changes are $ks$ is a lower step function and $ks$ is an upper step function for $kf.$

Exercise 7.4.5. One way is to mimic the proof that $f^{+}$ is integrable. Another is to use that $f^{-}=f^{+}-f.$

Exercise 7.4.6. Prove and use $|f|=f^{+}+f^{-}.$

Exercise 7.4.7. Integrate the inequalities $-|f|\leq f\leq |f|.$

Exercise 7.4.9. By the theorem both $\int _{a}^{c}f$ and $\int _{c}^{b}f$ exists. Let $\varepsilon>0$ be given. Since $f$ is integrable on $[a,c]$ there is a lower step function $S_{1}$ for $f$ on $[a,c]$ and an upper step function $S_{1}$ for $f$ on $[a,c],$ such that $\sum S_{1}-\sum s_{1}<\varepsilon /2.$ Similarly, there is a lower step function $S_{2}$ for $f$ on $[c,b]$ and an upper step function $S_{2}$ for $f$ on $[c,b],$ such that $\sum S_{2}-\sum s_{2}<\varepsilon /2.$ If $S_{1}=\sum _{k=1}^{m}M_{k}\mathbb {1}_{]x_{k-1},x_{k}[},$ $S_{1}=\sum _{k=1}^{m}M_{k}\mathbb {1}_{]x_{k-1},x_{k}[},$ $S_{2}=\sum _{k=m+1}^{n}M_{k}\mathbb {1}_{]x_{k-1},x_{k}[},$ and $S_{2}=\sum _{k=m+1}^{n}M_{k}\mathbb {1}_{]x_{k-1},x_{k}[},$ then $S:=\sum _{k=1}^{n}M_{k}\mathbb {1}_{]x_{k-1},x_{k}[}$ is a lower step function for $f$ on $[a,b]$ and $S:=\sum _{k=1}^{n}M_{k}\mathbb {1}_{]x_{k-1},x_{k}[}$ is an upper step function for $f$ on $[a,b].$

The rest is similar to the last part of the proof that the sum of two integrable functions is integrable. We include some of the details. $\sum S-\sum s=\left (\sum S_{1}+\sum S_{2}\right )-\left (\sum s_{1}+\sum s_{2}\right )=\left (\sum S_{1}-\sum s_{1}\right )+\left (\sum S_{2}-\sum s_{2}\right )<\varepsilon .$ It follows that $\int _{a}^{b}f$ and $\int _{a}^{c}f+\int _{c}^{b}f$ both are in the interval $\left [\sum s,\sum S\right ].$ Since this interval has length $<\varepsilon$ we have $\left |\int _{a}^{b}f-\left (\!\!\int _{a}^{c}f+\int _{c}^{b}f\!\!\right )\right |<\varepsilon .$ But $\varepsilon>0$ is arbitrary, consequently $\int _{a}^{b}f-\left (\!\!\int _{a}^{c}f+\int _{c}^{b}f\!\!\right ).$

Exercise 7.4.10. Similar to Exercise 7.4.9.

Exercise 7.4.11. Let $M$ be an upper bound for $f.$ Let $S$ a lower step function for $f$ with sum $\sum s=\sum m_{i}(x_{i}-x_{i-1})$ and $S$ be an upper step function for $f$ with sum $\sum S=\sum M_{i}(x_{i}-x_{i-1}),$ such that $\sum S-\sum s<\frac {\varepsilon }{2M}.$ We may assume $0\leq m_{i}$ and $M_{i}\leq M$ for all $i.$ Then $S^{2}$ is a lower step function for $f^{2},$ $S^{2}$ is an upper step function for $f^{2},$ and $\sum S^{2}-\sum s^{2}<\varepsilon .$

Exercise 7.4.12. For some $K,$ $f+K\geq 0$ and $f^{2}=(f+K)^{2}-2Kf-K^{2}.$

Exercise 7.5.2. Similar to the proof in the text, except we need to pay attention to signs.

Exercise 7.5.4. Similar to the proof in the text.

Exercise 7.6.1. Use FTC-Evaluation to calculate $\int _{1}^{b}f,$ then take the limit as $b\to \infty .$

Exercise 7.6.2. Similar to Exercise 7.6.1.

Exercise 7.7.1. Consider the real and imaginary parts.

Exercise 7.7.3. We may assume $y<x.$ Then squaring $\sqrt {x}-\sqrt {y}\leq \sqrt {x-y}$ and simplifying leads to $\sqrt {y}<\sqrt {x}.$ Rewriting this as $y<x$ implies $\sqrt {y}<\sqrt {x}$, etc., ending with $\sqrt {x}-\sqrt {y}\leq \sqrt {x-y}$ proves the inequality.

Exercise 7.7.6. This is a simple proof by induction. When $n=1,$ (7.11) is an equality. Suppose

$$ \sqrt{\left(x_{1}+x_{2}\right)^{2}+\left(y_{1}+y_{2}^{2}\right)}\leq\sqrt{\left(x_{1}\right)^{2}+\left(y_{1}\right)^{2}}+\sqrt{\left(x_{2}\right)^{2}+\left(y_{2}\right)^{2}}, $$

that is (7.11) holds with $n=2.$ Then

$$ \begin{aligned} & \sqrt{\left(\left(\sum_{i=1}^{n}\alpha_{i}\right)+\alpha_{n+1}\right)^{2}+\left(\left(\sum_{i=1}^{n}\beta_{i}\right)+\beta_{n+1}\right)^{2}}\\ & =\sqrt{\left(x_{1}+x_{2}\right)^{2}+\left(y_{1}+y_{2}^{2}\right)}\\ & \leq\sqrt{\left(x_{1}\right)^{2}+\left(y_{1}\right)^{2}}+\sqrt{\left(x_{2}\right)^{2}+\left(y_{2}\right)^{2}}\\ & =\sqrt{\left(\sum_{i=1}^{n}\alpha_{i}\right)^{2}+\left(\sum_{i=1}^{n}\beta_{i}\right)^{2}}+\sqrt{\left(\alpha_{n+1}\right)^{2}+\left(\beta_{n+1}\right)^{2}}. \end{aligned} $$

Hence, (7.11) follows by induction, if it holds for $n=2.$