Chapter 6

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 \(P_{e}\) of \([a, b]\) such that $$ r-\epsilon

Let \(f:[0,3] \rightarrow \mathbb{R}\) be defined by $$ f(x):=\left\\{\begin{array}{ll} 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 \(f:[a, b] \rightarrow \mathbb{R}\) 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 \(x_{0}\).

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 \(6.22\) repeatedly.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable and let \(F:[a, b] \rightarrow \mathbb{R}\) be given by \(F(x):=\) \(\int_{a}^{x} f(t) d t\) for \(x \in[a, b] .\) If \(f\) is nonnegative on \([a, b]\), then show that \(F\) is monotonically increasing on \([a, b] .\) Also, if \(f\) is monotonically increasing on \([a, b]\), then show that \(F\) is convex on \([a, b] .\) (Hint: To prove the convexity of \(F\), note that \(\left(F(x)-F\left(x_{1}\right)\right) /\left(x-x_{1}\right) \leq f(x) \leq\left(F\left(x_{2}\right)-F(x)\right) /\left(x_{2}-x\right)\) whenever \(a \leq x_{1}

Let \(f:[a, \infty) \rightarrow \mathbb{R}\) be a bounded function that 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 \(F\) satisfies a Lipschitz condition on \([a, \infty)\), and so \(F\) 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 (i) of Proposition 6.24.)

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

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable. Define \(F:[a, b] \rightarrow \mathbb{R}\) by $$ F(x):=\int_{x}^{b} f(t) d t \quad \text { for } x \in[a, b] $$ Show that \(F\) is continuous on \([a, b]\). Further, show that if \(f\) is continuous at \(c \in[a, b]\), then \(F\) is differentiable at \(c\) and \(F^{\prime}(c)=-f(c)\). (Hint: Propositions 6.8, 6.22, and 6.24.)

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 \(F:[c, d] \rightarrow \mathbb{R}\) by $$ F(y):=\int_{a}^{g(y)} f(t) d t \quad \text { for } y \in[c, d] $$ If \(g\) is differentiable at \(y_{0} \in[c, d]\) and \(f\) is continuous at \(g\left(y_{0}\right)\), then show that \(F\) is differentiable at \(y_{0}\) and \(F^{\prime}\left(y_{0}\right)=f\left(g\left(y_{0}\right)\right) g^{\prime}\left(y_{0}\right)\).

Let \(f, g:[a, b] \rightarrow \mathbb{R}\) be such that \(f\) is differentiable, \(f^{\prime}\) is integrable, and \(g\) is continuous. If $$ G(x):=\int_{a}^{x} g(t) d t \quad \text { and } \quad \tilde{G}(x):=\int_{x}^{b} g(t) d t \quad \text { for } x \in[a, b] $$ then show that $$ \int_{a}^{b} f(x) g(x) d x=f(b) G(b)-\int_{a}^{b} f^{\prime}(x) G(x) d x=f(a) \widetilde{G}(a)+\int_{a}^{b} f^{\prime}(x) \widetilde{G}(x) d x $$ (Compare Proposition 6.28.)

(Leibniz Rule for Integrals) Let \(f\) be a continuous function on \([a, b]\) and let \(u, v\) be differentiable functions on \([c, d] .\) If the ranges of \(u\) and \(v\) are contained in \([a, b]\), prove that $$ \frac{d}{d x} \int_{u(x)}^{v(x)} f(t) d t=f(v(x)) \frac{d v}{d x}-f(u(x)) \frac{d u}{d x} $$

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 \(n, m \in \mathbb{N}\). Find \(\lim _{m \rightarrow \infty} \int_{0}^{1} \frac{x^{n}}{(1+x)^{m}} d x\) and \(\lim _{n \rightarrow \infty} \int_{0}^{1} \frac{x^{n}}{(1+x)^{m}} d x\).

Find \(\lim _{n \rightarrow \infty} \int_{0}^{1} \frac{n x^{n-1}}{1+x} d x .\) (Hint: Proposition 6.28.)

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

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 \(6.28\) and \(6.29 .\) )

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 \(f\).

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

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

Domain Additivity of Lower/Upper Riemann Integrals) Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is 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 \([c, b] .\) Prove that \(L(f)=\) \(L\left(f_{1}\right)+L\left(f_{2}\right)\) and \(U(f)=U\left(f_{1}\right)+U\left(f_{2}\right) .\) (Compare Proposition 6.8.)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable and let \(g:[m(f), M(f)] \rightarrow \mathbb{R}\) be continuous. Show that \(g \circ f:[a, b] \rightarrow \mathbb{R}\) is integrable. (Hint: Given \(\epsilon>0\), find \(\delta>0\) using the uniform continuity of \(g .\) There is a partition \(P\) of \([a, b]\) such that \(U(P, f)-L(P, f)<\delta^{2}\). Divide the summands in \(U(P, f)-L(P, f)\) into two parts depending on whether or not \(M_{i}(f)-m_{i}(f)<\delta .\) Use the Riemann condition for \(g \circ f .\) )

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 \(I_{m+n, 0}=1 /(m+n+1) .\) )

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 $$ 1-\frac{1}{3}\left(\begin{array}{l} n \\ 1 \end{array}\right)+\frac{1}{5}\left(\begin{array}{l} n \\ 2 \end{array}\right)-\frac{1}{7}\left(\begin{array}{l} n \\ 3 \end{array}\right)+\cdots+\frac{(-1)^{n}}{2 n+1}\left(\begin{array}{l} n \\ n \end{array}\right)=\frac{\left(2^{n} n !\right)^{2}}{(2 n+1) !} $$ (Hint: For \(n \geq 0\), let \(I_{n}\) denote the given integral. Then \(I_{0}=a\) and \(I_{n}=a^{2} I_{n-1}-\int_{0}^{a} x g_{n}(x) d x\), where \(g_{n}(x):=x\left(a^{2}-x^{2}\right)^{n-1}\) for \(n \in \mathbb{N} .\) Use Integration by Parts to obtain \(I_{n}=2 n a^{2} I_{n-1} /(2 n+1)\) for \(n \in \mathbb{N} .\) )

(Taylor Theorem with Integral Remainder) Let \(n\) be a nonnegative integer and let \(f:[a, b] \rightarrow \mathbb{R}\) be such that \(f^{\prime}, f^{\prime \prime}, \ldots, f^{(n)}\) exist on \([a, b]\) and further, \(f^{(n)}\) is continuously differentiable 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 integral remainder is equal to $$ \frac{(b-a)^{n+1}}{n !} \int_{0}^{1}(1-s)^{n} f^{(n+1)}(a+s(b-a)) d s $$ and that there is \(c \in[a, b]\) such that the integral remainder is equal to $$ \frac{f^{(n+1)}(c)}{(n+1) !}(b-a)^{n+1} $$ (Hint: Induction on \(n\), Integration by Parts, and IVP of \(f^{(n+1)}\) ) [Note: Unlike the Lagrange form of the remainder, the integral remainder does not involve an undetermined number \(c \in(a, b) .]\)

(Taylor 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 \(\left.4.25 .\right)\)

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable. If \(\phi:[\alpha, \beta] \rightarrow \mathbb{R}\) is a differentiable function such that \(\phi^{\prime}\) is integrable on \([\alpha, \beta]\), and \(\phi^{\prime}(t) \neq 0\) for every \(t \in(\alpha, \beta)\), then show that the function \((f \circ \phi)\left|\phi^{\prime}\right|:[\alpha, \beta] \rightarrow \mathbb{R}\) is integrable and $$ \int_{a}^{b} f(x) d x=\int_{\alpha}^{\beta} f(\phi(t))\left|\phi^{\prime}(t)\right| d t . $$ (Hint: Either \(\phi^{\prime}>0\) or \(\phi^{\prime}<0\) on \((\alpha, \beta)\). Let \(\psi:=(f \circ \phi)|\phi|\) and let \(\epsilon>0\) be given. For any partition \(P\) of \([a, b]\), obtain a partition \(Q\) of \([\alpha, \beta]\) with \(L(P, f)

(Bliss Theorem) Let \(f, g:[a, b] \rightarrow \mathbb{R}\) be any functions. 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 $$ \widetilde{S}_{n}:=\sum_{i=1}^{k_{n}} f\left(s_{n, i}\right) g\left(t_{n, i}\right)\left(x_{n, i}-x_{n, i-1}\right) . $$ Show that if \(f\) is integrable, \(g\) is continuous, and \(\mu\left(P_{n}\right) \rightarrow 0\), then $$ \tilde{S}_{n} \rightarrow \int_{a}^{b} f(x) g(x) d x $$ (Hint: If \(\mathcal{T}_{n}:=\left\\{s_{n, i}: i=1, \ldots, k_{n}\right\\}\), then \(S\left(P_{n}, \mathcal{T}_{n}, f g\right) \rightarrow \int_{a}^{b} f(x) g(x) d x\) Also, \(g\) is uniformly continuous.)

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 \(\operatorname{sum} \sum_{i=1}^{n} f\left(x_{i}\right)\left(G\left(x_{i}\right)-G\left(x_{i-1}\right)\right)\). 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 Exercise \(6.45\) and note \(m(g)(f(b)-f(a)) \leq\) \(\left.\sum_{i=1}^{n} G\left(x_{i-1}\right)\left(f\left(x_{i}\right)-f\left(x_{i-1}\right)\right) \leq M(g)(f(b)-f(a)) .\right)\)

(First Mean Value Theorem for Integrals) Let \(f:[a, b] \rightarrow \mathbb{R}\) be a continuous function and let \(g:[a, b] \rightarrow \mathbb{R}\) 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 \(8.72 .\) ]

(Second Mean Value Theorem for Integrals) Let \(f:[a, b] \rightarrow \mathbb{R}\) be a monotonic function and let \(g:[a, b] \rightarrow \mathbb{R}\) 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 6.46.)

(Cauchy Condition) Let \(f:[a, b] \rightarrow \mathbb{R}\) be a bounded function. Show that \(f\) is integrable on \([a, b]\) if and only if for every \(\epsilon>0\), there is a partition \(P\) of \([a, b]\) such that for all tag sets \(\mathcal{T}\) and \(\mathcal{T}^{\prime}\) associated with \(P\), $$ \left|S(P, \mathcal{T}, f)-S\left(P, \mathcal{T}^{\prime}, f\right)\right|<\epsilon $$ (Hint: Argue as in the proof of Proposition 6.36.)

Let \(E\) be a bounded subset of \(\mathbb{R}\) and let \(\partial E\) denote the boundary of \(E\). (i) Show that if \(E\) is of content zero, then \(\bar{E}:=E \cup \partial E\) is of content zero. (ii) Show that \(E\) is of content zero if and only if \(E\) has no interior points and \(\partial E\) is of content zero.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be integrable, and let \(g:[a, b] \rightarrow \mathbb{R}\) be a bounded function such that the set \(\\{x \in[a, b]: g(x) \neq f(x)\\}\) is of content zero. Show that \(g\) is integrable and $$ \int_{a}^{b} g(x) d x=\int_{a}^{b} f(x) d x $$ (Compare Proposition 6.13.)