Chapter 6

Let \(c \in[a, b]\) and \(f:[a, b] \rightarrow \mathbb{R}\) be given by $$ f(x):=\left\\{\begin{array}{ll} 0 & \text { if } a \leq x \leq c \\ 1 & \text { if } c

Let \(c \in(a, b)\) and \(f:[a, b] \rightarrow \mathbb{R}\) be given by $$ f(x):=\left\\{\begin{array}{ll} (x-c) /(a-c) & \text { if } a \leq x \leq c, \\ (x-c) /(b-c) & \text { if } c

Let \(f:[0,1] \rightarrow \mathbb{R}\) be given by $$ f(x):=\left\\{\begin{array}{ll} 1+x & \text { if } x \text { is rational } \\ 0 & \text { if } x \text { is irrational. } \end{array}\right. $$ Is \(f\) integrable?

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable. Show that the Riemann integral of \(f\) is the unique real number \(r\) satisfying the following condition: For every \(\epsilon>0\), there is a partition \(P_{\epsilon}\) of \([a, b]\) such that $$ r-\epsilon

Let \(f:[0,3] \rightarrow \mathbb{R}\) be defined by $$ f(x):=\left\\{\begin{aligned} 0 & \text { if } 0 \leq x \leq 1 \\ 2 & \text { if } 1

Let \(f, g:[a, b] \rightarrow \mathbb{R}\) be bounded functions. Show that $$ L(f)+L(g) \leq L(f+g) \quad \text { and } \quad U(f+g) \leq U(f)+U(g) $$ Hence conclude that if \(f\) and \(g\) are integrable, then so is \(f+g\), and the Riemann integral of \(f+g\) is equal to the sum of the Riemann integrals of \(f\) and \(g\).

Let \(f, g:[a, b] \rightarrow \mathbb{R}\) be integrable. Show that the functions \(\max (f, g)\) : \([a, b] \rightarrow \mathbb{R}\) and \(\min (f, g):[a, b] \rightarrow \mathbb{R}\) given by \(\max (f, g)(x)=\max \\{f(x), g(x)\\} \quad\) and \(\quad \min (f, g)(x)=\min \\{f(x), g(x)\\}\) are integrable. (Hint: \(\max (f, g)=(f+g+|f-g|) / 2\) and \(\min (f, g)=\) \((f+g-|f-g|) / 2 .)\)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a function. Show that \(f\) is integrable if (i) \(r f\) is integrable for some nonzero \(r \in \mathbb{R}\), or (ii) if \(f\) is bounded, \(f(x) \neq 0\) for all \(x \in[a, b]\), and \(1 / f\) is integrable.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be any function. Suppose there is \(r \in \mathbb{R}\) and for each \(n \in \mathbb{N}\), there are integrable functions \(g_{n}, h_{n}:[a, b] \rightarrow \mathbb{R}\) with \(g_{n} \leq f \leq h_{n}\) such that \(\int_{a}^{b} g_{n}(x) d x \rightarrow r\) and \(\int_{a}^{b} h_{n}(x) d x \rightarrow r\) as \(n \rightarrow \infty\). Show that \(f\) is integrable and the Riemann integral of \(f\) is equal to \(r\).

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable and \(f(x) \geq 0\) for all \(x \in[a, b] .\) Show that \(\int_{a}^{b} f(x) d x \geq 0 .\) If, in addition, \(f\) is continuous and \(\int_{a}^{b} f(x) d x=0\), then show that \(f(x)=0\) for all \(x \in[a, b] .\) Give an example of an integrable function on \([a, b]\) such that \(f(x) \geq 0\) for all \(x \in[a, b]\) and \(\int_{a}^{b} f(x) d x=0\), but \(f(x) \neq 0\) for some \(x \in[a, b]\).

Evaluate the following limits. (i) \(\lim _{h \rightarrow 0} \frac{1}{h} \int_{x}^{x+h} \frac{d u}{u+\sqrt{u^{2}+1}}\), (ii) \(\lim _{x \rightarrow 0} \frac{1}{x^{3}} \int_{0}^{x} \frac{t^{2} d t}{t^{4}+1}\), (iii) \(\lim _{x \rightarrow 0} \frac{1}{x^{6}} \int_{0}^{x^{2}} \frac{t^{2} d t}{t^{6}+1}\), (iv) \(\lim _{x \rightarrow x_{0}} \frac{x}{x-x_{0}} \int_{x_{0}}^{x} f(t) d t\) (v) \(\lim _{x \rightarrow x_{0}} \frac{x}{x^{2}-x_{0}^{2}} \int_{x_{0}}^{x} f(t) d t\), provided \(f\) is continuous at \(x_{0}\).

If \(x:=\int_{0}^{y} \frac{d t}{\sqrt{1+t^{2}}}\), find \(\frac{d^{2} y}{d x^{2}}\)

Let \(a, b, c \in \mathbb{R}\) with \(a

Let \(n \in \mathbb{N}\). Find a function \(f:[-1,1] \rightarrow \mathbb{R}\) for which \(f^{(n)}(0)\) exists, but \(f^{(n+1)}(0)\) does not. (Hint: Begin with the absolute value function and use part (ii) of Proposition \(6.20\) repeatedly.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be continuous and consider the function \(F:[a, b] \rightarrow \mathbb{R}\) given by \(F(x):=\int_{a}^{x} f(t) d t\) for \(x \in[a, b]\). If \(f(x) \geq 0\) for all \(x \in[a, b]\), then show that \(F\) is monotonically increasing on \([a, b]\), and if \(f\) monotonically increasing on \([a, b]\), then \(F\) is convex on \([a, b]\). (Hint: Part (i) of Proposition \(4.27\) and Part (i) of Proposition 4.31.)

Let \(f:[a, \infty) \rightarrow \mathbb{R}\) be a bounded function such that \(f\) is integrable on \([a, x]\) for every \(x \geq a\). Let \(F(x):=\int_{a}^{x} f(t) d t\) for \(x \geq a\). Show that \(F\) is uniformly continuous on \([a, \infty)\).

Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be continuous and \(f(x) \geq 0\) for all \(x \in[0, \infty)\). If for each \(b>0\), the area bounded by the \(x\) -axis, the lines \(x=0, x=b\), and the curve \(y=f(x)\) is given by \(\sqrt{b^{2}+1}-1\), determine the function \(f\).

Let \(p\) be a real number and let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function such that \(f(x+p)=f(x)\) for all \(x \in \mathbb{R}\). (Such a function is said to be periodic.) Show that the integral \(\int_{a}^{a+p} f(t) d t\) has the same value for every real number \(a\). (Hint: Part (ii) of Proposition 6.21.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be continuous. Show that for every \(x \in[a, b]\), $$ \int_{a}^{x}\left[\int_{a}^{u} f(t) d t\right] d u=\int_{a}^{x}(x-u) f(u) d u . $$

2Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable. Define \(G:[a, b] \rightarrow \mathbb{R}\) by $$ G(x):=\int_{x}^{b} f(t) d t . $$ Show that \(G\) is continuous on \([a, b] .\) Further, show that if \(f\) is continuous at \(c \in[a, b]\), then \(G\) is differentiable at \(c\) and \(G^{\prime}(c)=-f(c)\). (Hint: Propositions \(6.7,6.20\), and \(6.21\).)

Let \(g:[c, d] \rightarrow \mathbb{R}\) be such that \(g([c, d]) \subseteq[a, b]\), and let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable. Define \(G:[c, d] \rightarrow \mathbb{R}\) by $$ G(y):=\int_{a}^{g(y)} f(t) d t $$ If \(g\) is differentiable at \(y_{0} \in[c, d]\) and \(f\) is continuous at \(g\left(y_{0}\right)\), then show that \(G\) is differentiable at \(y_{0}\) and \(G^{\prime}\left(y_{0}\right)=f\left(g\left(y_{0}\right)\right) g^{\prime}\left(y_{0}\right)\).

(Leibniz's Rule for Integrals) Let \(f\) be a continuous function on \([a, b]\) and \(u, v\) be differentiable functions on \([c, d] .\) If the ranges of \(u\) and \(v\) are contained in \([a, b]\), prove that $$ \frac{d}{d x} \int_{u(x)}^{v(x)} f(t) d t=\left[f(v(x)) \frac{d v}{d x}-f(u(x)) \frac{d u}{d x}\right] $$

For \(x \in \mathbb{R}\), let \(F(x):=\int_{1}^{2 x} \frac{1}{1+t^{2}} d t\) and \(G(x):=\int_{0}^{x^{2}} \frac{1}{1+\sqrt{|t|}} d t\). Find \(F^{\prime}\) and \(G^{\prime}\).

Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be continuous. Find \(f(2)\) if for all \(x \geq 0\), (i) \(\int_{0}^{x} f(t) d t=x^{2}(1+x)\), (ii) \(\int_{0}^{f(x)} t^{2} d t=x^{2}(1+x)\), (iii) \(\int_{0}^{x^{2}} f(t) d t=x^{2}(1+x)\), (iv) \(\int_{0}^{x^{2}(1+x)} f(t) d x=x\).

Find \(\lim _{n \rightarrow \infty} \int_{0}^{1} \frac{n x^{n-1}}{1+x} d x .\) (Hint: Proposition 6.25.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a differentiable function. If \(F\) is an antiderivative of \(f\) on \([a, b]\), then show that $$ \int_{a}^{b} f^{2}(x) d x=F(b) F^{\prime}(b)-F(a) F^{\prime}(a)-\int_{a}^{b} F(x) F^{\prime \prime}(x) d x $$

Evaluate (i) \(\int_{0}^{1 / 4} \frac{x}{\sqrt{1-4 x^{2}}} d x\), (ii) \(\int_{1}^{8} x^{1 / 3}\left(x^{4 / 3}-1\right)^{1 / 2} d x\). (Hint: Proposition \(6.26 .\) )

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a differentiable function such that \(f^{\prime}\) is continuous on \([a, b]\) and \(f^{\prime}(x) \neq 0\) for all \(x \in[a, b]\). If \(f([a, b])=[c, d]\), then show that \(f^{-1}:[c, d] \rightarrow \mathbb{R}\) is integrable and $$ \int_{c}^{d} f^{-1}(y) d y=f^{-1}(d) d-f^{-1}(c) c-\int_{f^{-1}(c)}^{f^{-1}(d)} f(x) d x $$ (Hint: Propositions \(6.25\) and \(6.26 .\) )

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function and define \(g:[-b,-a] \rightarrow \mathbb{R}\) by \(g(t):=f(-t) .\) Show that \(L(g)=L(f)\) and \(U(g)=U(f) .\) Deduce that \(g\) is integrable on \([-b,-a]\) if and only if \(f\) is integrable on \([a, b]\) and in that case the Riemann integral of \(g\) is equal to the Riemann integral of \(f\). (Compare the proof of part (ii) of Proposition \(6.26 .\) )

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable and for \(n \in \mathbb{N}\), let \(P_{n}\) be a partition of \([a, b]\) such that \(U\left(P_{n}, f\right)-L\left(P_{n}, f\right) \rightarrow 0 .\) Show that \(U\left(P_{n}, f\right) \rightarrow \int_{a}^{b} f(x) d x\) \(L\left(P_{n}, f\right) \rightarrow \int_{a}^{b} f(x) d x\), and also \(S\left(P_{n}, f\right) \rightarrow \int_{a}^{b} f(x) d x\), where \(S\left(P_{n}, f\right)\) is a Riemann sum for \(f\) corresponding to \(P_{n} .\) (Compare Proposition \(6.5\) and Lemma 6.30.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be an integrable function. If \(\left(P_{n}\right)\) is a sequence of partitions of \([a, b]\) such that \(\mu\left(P_{n}\right) \rightarrow 0\), then show that \(U\left(P_{n}, f\right)-L\left(P_{n}, f\right) \rightarrow\) 0. Is the converse true?

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function. Without using Lemma \(6.30\), show that \(f\) is Riemann integrable if and only if there is \(r \in \mathbb{R}\) satisfying the following condition: Given \(\epsilon>0\), there is a partition \(P_{\epsilon}\) of \([a, b]\) such that \(|S(P, f)-r|<\epsilon\), where \(P\) is any refinement of \(P_{\epsilon}\) and \(S(P, f)\) is any Riemann sum for \(f\) corresponding to \(P\).

Assuming that \(f\) is integrable on \([0,1]\), show that $$ \lim _{n \rightarrow \infty} \frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n}{n}\right)\right]=\int_{0}^{1} f(x) d x . $$

Consider the sequence whose \(n\) th term is given by the following. In each case, determine the limit of the sequence by expressing the \(n\) th term as a Riemann sum for a suitable function. (i) \(\frac{1}{n^{17}} \sum_{i=1}^{n} i^{16}\), (ii) \(\frac{1}{n^{5 / 2}} \sum_{i=1}^{n} i^{3 / 2}\), (iii) \(\sum_{i=1}^{n} \frac{1}{\sqrt{i n+n^{2}}}\), (iv) \(\frac{1}{n}\left\\{\sum_{i=1}^{n}\left(\frac{i}{n}\right)+\sum_{i=n+1}^{2 n}\left(\frac{i}{n}\right)^{3 / 2}+\sum_{i=2 n+1}^{3 n}\left(\frac{i}{n}\right)^{2}\right\\}\).

Do \(\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{i+n}}\) and \(\lim _{n \rightarrow \infty} \frac{1}{n^{18}} \sum_{i=1}^{n} i^{16}\) exist? If yes, find them.

Find an approximate value of \(1^{1 / 3}+2^{1 / 3}+\cdots+1000^{1 / 3}\).

Let \(a, b \in \mathbb{R}\) with \(0 \leq a

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function. For \(c \in(a, b)\), let \(f_{1}\) and \(f_{2}\) denote the restrictions of \(f\) to the subintervals \([a, c]\) and \([c, b]\) respectively. Prove the following: (i) \(L(f)=L\left(f_{1}\right)+L\left(f_{2}\right)\), (ii) \(U(f)=U\left(f_{1}\right)+U\left(f_{2}\right)\). [Note: The results in (i) and (ii) are refined versions of Proposition \(6.7\), and may be referred to as Domain Additivity of Lower Riemann Integrals and Domain Additivity of Upper Riemann Integrals, respectively.]

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable and \(\phi:[m(f), M(f)] \rightarrow \mathbb{R}\) be continuous. Show that \(\phi \circ f:[a, b] \rightarrow \mathbb{R}\) is integrable. (Hint: Given \(\epsilon>0\), find \(\delta>0\) using the uniform continuity of \(\phi\). There is a partition \(P\) of \([a, b]\) such that \(U(P, f)-L(P, f)<\delta^{2}\). Divide the sum in \(U(P, f)-L(P, f)\) into two classes depending on whether \(M_{i}(f)-m_{i}(f)\) is less than \(\delta\), or greater than or equal to \(\delta\). Use the Riemann condition for \(\phi \circ f\).)

Let \(f_{1}, \ldots, f_{m}:[a, b] \rightarrow \mathbb{R}\) be integrable functions and let \(r_{j}:=\int_{a}^{b} f_{j}(x) d x\) for \(j=1, \ldots, m .\) Show that the function \(\sqrt{f_{1}^{2}+\cdots+f_{m}^{2}}\) is integrable and $$ \sqrt{r_{1}^{2}+\cdots+r_{m}^{2}} \leq \int_{a}^{b} \sqrt{f_{1}^{2}(x)+\cdots+f_{m}^{2}(x)} d x $$ (Hint: Note that \(\sum_{j=1}^{m} r_{j}^{2}=\sum_{j=1}^{m} r_{j} \int_{a}^{b} f_{j}(x) d x=\int_{a}^{b}\left(\sum_{j=1}^{m} r_{j} f_{j}(x)\right) d x\) and use Proposition 1.12.)

Let \(m, n \in \mathbb{Z}\) with \(m, n \geq 0 .\) Show that $$ \int_{0}^{1} x^{m}(1-x)^{n} d x=\frac{m ! n !}{(m+n+1) !} $$ (Hint: If \(n \in \mathbb{N}\) and \(I_{m, n}\) denotes the given integral, then using Integration by Parts, \(I_{m, n}=[n /(m+1)] I_{m+1, n-1}\), and \(\left.I_{m+n, 0}=1 /(m+n+1) .\right)\)

Let \(a \in \mathbb{R}\) and \(n \in \mathbb{Z}\) with \(n \geq 0 .\) Show that $$ \int_{0}^{a}\left(a^{2}-x^{2}\right)^{n} d x=\frac{\left(2^{n} n !\right)^{2}}{(2 n+1) !} \cdot a^{2 n+1} . $$ Deduce that $$ 1-\frac{1}{3}\left(\begin{array}{l} n \\ 1 \end{array}\right)+\frac{1}{5}\left(\begin{array}{l} n \\ 2 \end{array}\right)-\frac{1}{7}\left(\begin{array}{c} n \\ 3 \end{array}\right)+\cdots+\frac{(-1)^{n}}{2 n+1}\left(\begin{array}{l} n \\ n \end{array}\right)=\frac{\left(2^{n} n !\right)^{2}}{(2 n+1) !} $$ (Hint: If \(n \in \mathbb{N}\) and \(I_{n}\) denotes the given integral, then \(I_{n}=a^{2} I_{n-1}-\) \(\int_{0}^{a} x\left[x\left(a^{2}-x^{2}\right)^{n-1}\right] d x\), and using Integration by Parts, \(I_{n}=a^{2}[2 n /(2 n+\) 1) \(] I_{n-1}\), and \(I_{0}=a .\) )

(Taylor's Theorem with Integral Remainder) Let \(n\) be a nonnegative integer and let \(f:[a, b] \rightarrow \mathbb{R}\) be such that \(f^{\prime}, f^{\prime \prime}, \ldots, f^{(n+1)}\) exist and \(f^{(n+1)}\) is continuous on \([a, b]\). Show that $$ f(b)=f(a)+f^{\prime}(a)(b-a)+\cdots+\frac{f^{(n)}(a)}{n !}(b-a)^{n}+\frac{1}{n !} \int_{a}^{b}(b-t)^{n} f^{(n+1)}(t) d t $$ Further, show that the remainder is equal to $$ \frac{(b-a)^{n+1}}{n !} \int_{0}^{1}(1-s)^{n} f^{(n+1)}(a+s(b-a)) d s $$ (Hint: Induction on \(n\) and Integration by Parts.) [Note: The integral remainder does not involve an undetermined number \(c \in(a, b) .]\)

(Taylor's Theorem for Integrals) Let \(n \in \mathbb{N}\) and \(f:[a, b] \rightarrow \mathbb{R}\) be such that \(f^{\prime}, f^{\prime \prime}, \ldots, f^{(n-1)}\) exist on \([a, b]\), and further, \(f^{(n-1)}\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Show that there is \(c \in(a, b)\) such that \(\int_{a}^{b} f(x) d x=f(a)(b-a)+\cdots+\frac{f^{(n-1)}(a)}{n !}(b-a)^{n}+\frac{f^{(n)}(c)}{(n+1) !}(b-a)^{n+1} .\) (Hint: For \(x \in[a, b]\), define \(F(x):=\int_{a}^{x} f(t) d t\) and apply Proposition 4.23.)

(Theorem of Bliss) Let \(f, g:[a, b] \rightarrow \mathbb{R}\) be integrable. For each \(n \in \mathbb{N}\), consider a partition \(P_{n}:=\left\\{x_{n, 0}, x_{n, 1}, \ldots, x_{n, k_{n}}\right\\}\) of \([a, b]\), and for \(i=\) \(1, \ldots, k_{n}\), let \(s_{n, i}, t_{n, i} \in\left[x_{n, i-1}, x_{n, i}\right]\), and let $$ \widetilde{S}\left(P_{n}, f g\right):=\sum_{i=1}^{k_{n}} f\left(s_{n, i}\right) g\left(t_{n, i}\right)\left(x_{n, i}-x_{n, i-1}\right) $$

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a monotonic function. If \(G:[a, b] \rightarrow \mathbb{R}\) is differentiable and \(G^{\prime}\) is continuous, then show that there is \(c \in[a, b]\) such that $$ \int_{a}^{b} f(x) G^{\prime}(x) d x=f(b) G(b)-f(a) G(a)-G(c)[f(b)-f(a)] $$ (Hint: Given any partition \(P=\left\\{x_{0}, x_{1}, \ldots, x_{n}\right\\}\) of \([a, b]\), consider the sum \(\sum_{i=1}^{n} f\left(x_{i}\right)\left[G\left(x_{i}\right)-G\left(x_{i-1}\right)\right] .\) Write it as \(f(b) G(b)-f(a) G(a)-\) \(\sum_{i=1}^{n} G\left(x_{i-1}\right)\left[f\left(x_{i}\right)-f\left(x_{i-1}\right)\right]\) and also as \(\sum_{i=1}^{n} f\left(x_{i}\right) G^{\prime}\left(s_{i}\right)\left(x_{i}-x_{i-1}\right)\) for some \(s_{i} \in\left[x_{i-1}, x_{i}\right] .\) Use the Theorem of Bliss (Exercise 48) and the inequalities \(m(g)[f(b)-f(a)] \leq \sum_{i=1}^{n} G\left(x_{i-1}\right)\left[f\left(x_{i}\right)-f\left(x_{i-1}\right)\right] \leq\) \(M(g)[f(b)-f(a)] .)\)

(First Mean Value Theorem for Integrals) Let \(f:[a, b] \rightarrow \mathbb{R}\) be a continuous function and \(g:[a, b] \rightarrow \mathbb{R}\) be a nonnegative integrable function. Use the IVP of \(f\) to show that there is \(c \in[a, b]\) such that $$ \int_{a}^{b} f(x) g(x) d x=f(c) \int_{a}^{b} g(x) d x $$ Give examples to show that neither the continuity of \(f\) nor the nonnegativity of \(g\) can be omitted. [Note: For another version of this result, see Exercise 72 .

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a monotonic function and \(g:[a, b] \rightarrow \mathbb{R}\) be either a nonnegative integrable function or a continuous function. Show that there is \(c \in[a, b]\) such that $$ \int_{a}^{b} f(x) g(x) d x=f(a) \int_{a}^{c} g(x) d x+f(b) \int_{c}^{b} g(x) d x $$ Give an example to show that the monotonicity of \(f\) cannot be omitted. (Hint: Without loss of generality, suppose \(f\) is (monotonically) increasing. Let \(G(x):=\int_{a}^{x} g(t) d t\) for \(x \in[a, b] .\) If \(g\) is a nonnegative integrable function, then \(f(a) G(b) \leq \int_{a}^{b} f(x) g(x) d x \leq f(b) G(b)\). If \(g\) is continuous, use Exercise 49.)

Let \(D\) be a bounded subset of \(\mathbb{R}\) and \(f: D \rightarrow \mathbb{R}\) be a bounded function. Suppose \(D \subseteq[a, b]\) for \(a, b \in \mathbb{R}\) and \(f^{*}:[a, b] \rightarrow \mathbb{R}\) is defined by $$f^{*}(x):=\left\\{\begin{array}{ll} f(x) & \text { if } x \in D \\ 0 & \text { otherwise } \end{array}\right.$$ The function \(f\) is said to be integrable (on \(D)\) if the function \(f^{*}\) is integrable (on \([a, b])\). In this case, we define the Riemann integral of \(f\) \((\) on \(D)\) by $$ \int_{D} f(x) d x:=\int_{a}^{b} f^{*}(x) d x $$ (i) Show that the above definition is independent of the interval \([a, b]\) containing \(D\). (ii) Show that analogues of Propositions \(6.15\) and \(6.18\) hold for integrable functions on \(D\).

A bounded subset \(E\) of \(\mathbb{R}\) is said to be of (one-dimensional) content zero if the following condition holds: For every \(\epsilon>0\), there is a finite number of closed intervals whose union contains \(E\) and the sum of whose lengths is less than \(\epsilon\). Prove the following statements: (i) A subset of a set of content zero is of content zero. (ii) A finite union of sets of content zero is of content zero. (iii) If \(E\) is of content zero and \(\partial E\) denotes the boundary of \(E\), then \(E \cup \partial E\) is of content zero. (iv) A set \(E\) is of content zero if and only if the interior of \(E\) is empty and \(\partial E\) is of content zero. (v) Every finite subset of \(\mathbb{R}\) is of content zero. (vi) The infinite set \(\\{1 / n: n \in \mathbb{N}\\}\) is of content zero. (vii) The infinite set \(\mathbb{Q} \cap[0,1]\) is not of content zero.