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A Course in Calculus and Real Analysis

Sudhir R. Ghorpade, Balmohan V. Limaye

Chapter 6

Integration - all with Video Answers

Educators

Chapter Questions

02:08

Problem 1

Let $c \in[a, b]$ and $f:[a, b] \rightarrow \mathbb{R}$ be given by
$$
f(x):=\left\{\begin{array}{ll}
0 & \text { if } a \leq x \leq c \\
1 & \text { if } c<x \leq b
\end{array}\right.
$$
Show from first principles that $f$ is integrable on $[a, b] .$ Also, prove that this follows from Proposition $6.10$.

Regina Hays
Regina Hays
Numerade Educator
02:59

Problem 2

Let $c \in(a, b)$ and $f:[a, b] \rightarrow \mathbb{R}$ be given by
$$
f(x):=\left\{\begin{array}{ll}
(x-c) /(a-c) & \text { if } a \leq x \leq c, \\
(x-c) /(b-c) & \text { if } c<x \leq b
\end{array}\right.
$$
Show from first principles that $f$ is integrable on $[a, b] .$ Also, prove that this follows from Proposition $6.10$. (Hint: For $n \in \mathbb{N}$, consider the partition $P_{n}:=\{a, a+(c-a) / n, \ldots, a+(c-a)(n-1) / n, c, c+(b-c) / n, \ldots$
$c+(b-c)(n-1) / n, b\} .)$

Nick Johnson
Nick Johnson
Numerade Educator
03:05

Problem 3

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?

Linda Hand
Linda Hand
Numerade Educator
02:59

Problem 4

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
$$
r-\epsilon<L\left(P_{\epsilon}, f\right) \leq r \leq U\left(P_{\epsilon}, f\right)<r+\epsilon
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:30

Problem 5

Let $f:[0,3] \rightarrow \mathbb{R}$ be defined by
$$
f(x):=\left\{\begin{aligned}
0 & \text { if } 0 \leq x \leq 1 \\
2 & \text { if } 1<x \leq 2 \\
-1 & \text { if } 2<x \leq 3
\end{aligned}\right.
$$
Show that $f$ is neither monotonic nor continuous on $[0,3]$, but $f$ is integrable on $[0,3] .$ Find the Riemann integral of $f$.

Lucas Finney
Lucas Finney
Numerade Educator
02:08

Problem 6

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 $g$.

Regina Hays
Regina Hays
Numerade Educator
02:19

Problem 7

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 $\min (f, g)=$ $(f+g-|f-g|) / 2 .)$

Regina Hays
Regina Hays
Numerade Educator
01:30

Problem 8

Give examples of bounded functions $f, g:[a, b] \rightarrow \mathbb{R}$ that are not integrable, but $|f|, f+g$, and $f g$ are all integrable.

Gregory Higby
Gregory Higby
Numerade Educator
02:46

Problem 9

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.

Nick Johnson
Nick Johnson
Numerade Educator
02:46

Problem 10

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 $r$.

Nick Johnson
Nick Johnson
Numerade Educator
02:46

Problem 11

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 $x \in[a, b]$.

Nick Johnson
Nick Johnson
Numerade Educator
03:04

Problem 12

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 $x_{0}$.

Foster Wisusik
Foster Wisusik
Numerade Educator
00:53

Problem 13

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

Doruk Isik
Doruk Isik
Numerade Educator
02:09

Problem 14

Let $a, b, c \in \mathbb{R}$ with $a<c<b$ and for $j=1,2,3$, consider $f_{j}:[a, b] \rightarrow \mathbb{R}$ given by
(i) $f_{1}(x):=\left\{\begin{array}{ll}0 & \text { if } x \leq c, \\ 1 & \text { if } c<x, \\ 0 & \text { if } x \neq c, \\ 1 & \text { if } x=c,\end{array}\right.$
(iii) $f_{3}(x):=\left\{\begin{array}{ll}(x-c) /(a-c) & \text { if } x \leq c \\ (x-c) /(b-c) & \text { if } c<x\end{array}\right.$
For $j=1,2,3$, let $F_{j}(x):=\int_{a}^{x} f_{j}(t) d t, x \in[a, b]$. Find $F_{j}$ for $j=1,2,3$. Verify that $f_{1}$ is discontinuous at $c, F_{1}$ is continuous but not differentiable at $c, f_{2}$ is discontinuous at $c, F_{2}$ is differentiable at $c$ but $F_{2}^{\prime}(c) \neq f_{2}(c)$, $f_{3}$ is continuous at $c$ but not differentiable at $c, F_{3}$ is differentiable at $c$ and $F_{3}^{\prime}(c)=f_{3}(c)$

Aman Gupta
Aman Gupta
Numerade Educator
01:17

Problem 15

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.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:18

Problem 16

If $f:[a, b] \rightarrow \mathbb{R}$ is continuous, then prove part (i) of the FTC using part (ii) of the FTC. (Hint: Two antiderivatives of $f$ differ by an additive constant.)

Bobby Barnes
Bobby Barnes
University of North Texas
01:31

Problem 17

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.)

Stark Ledbetter
Stark Ledbetter
Numerade Educator
09:39

Problem 18

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 $[a, \infty)$.

Luuk Verhoeven
Luuk Verhoeven
Numerade Educator
04:49

Problem 19

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 $f$.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
03:30

Problem 20

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.)

Satyam Gupta
Satyam Gupta
Numerade Educator
01:29

Problem 21

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

Adrian Co
Adrian Co
Numerade Educator
02:46

Problem 22

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 $6.21$.)

Nick Johnson
Nick Johnson
Numerade Educator
01:46

Problem 23

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 $G^{\prime}\left(y_{0}\right)=f\left(g\left(y_{0}\right)\right) g^{\prime}\left(y_{0}\right)$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:13

Problem 24

(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
$$
\frac{d}{d x} \int_{u(x)}^{v(x)} f(t) d t=\left[f(v(x)) \frac{d v}{d x}-f(u(x)) \frac{d u}{d x}\right]
$$

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
17:51

Problem 25

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 $G^{\prime}$.

CG
Coleman Green
Numerade Educator
00:44

Problem 26

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) $\int_{0}^{x^{2}(1+x)} f(t) d x=x$.

Gio Maya
Gio Maya
Numerade Educator
07:25

Problem 27

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

Leon Druch
Leon Druch
Numerade Educator
02:29

Problem 28

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

Joseph David
Joseph David
Numerade Educator
03:04

Problem 29

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

Sirat Shah
Sirat Shah
Numerade Educator
01:07

Problem 30

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 $6.26 .$ )

Tyler Moulton
Tyler Moulton
Numerade Educator
02:46

Problem 31

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 $6.26 .$ )

Nick Johnson
Nick Johnson
Numerade Educator
02:46

Problem 32

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 $6.26 .$ )

Nick Johnson
Nick Johnson
Numerade Educator
02:59

Problem 33

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.)

Nick Johnson
Nick Johnson
Numerade Educator
02:59

Problem 34

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?

Nick Johnson
Nick Johnson
Numerade Educator
02:59

Problem 35

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 $P$.

Nick Johnson
Nick Johnson
Numerade Educator
05:24

Problem 36

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

Joseph David
Joseph David
Numerade Educator
View

Problem 37

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) $\frac{1}{n}\left\{\sum_{i=1}^{n}\left(\frac{i}{n}\right)+\sum_{i=n+1}^{2 n}\left(\frac{i}{n}\right)^{3 / 2}+\sum_{i=2 n+1}^{3 n}\left(\frac{i}{n}\right)^{2}\right\}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:13

Problem 38

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.

Nick Johnson
Nick Johnson
Numerade Educator
02:19

Problem 39

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

Ziya Ogron
Ziya Ogron
Numerade Educator
01:28

Problem 40

Let $a, b \in \mathbb{R}$ with $0 \leq a<b$ and $m \in \mathbb{N}$, and let $f:[a, b] \rightarrow \mathbb{R}$ be defined by $f(x):=x^{m}$. Show from the first principles that
$$
\int_{a}^{b} f(x) d x=\frac{b^{m+1}-a^{m+1}}{m+1} .
$$
(Hint: If $P=\left\{x_{0}, x_{1}, \ldots, x_{n}\right\}$ is a partition of $[a, b]$, then for each $j=$ $0,1, \ldots, m$, we have $L(P, f) \leq \sum_{i=1}^{n} x_{i}^{m-j} x_{i-1}^{j}\left(x_{i}-x_{i-1}\right) \leq U(P, f)$. Also,
$\left.\sum_{j=0}^{m}\left[\sum_{i=1}^{n} x_{i}^{m-j} x_{i-1}^{j}\left(x_{i}-x_{i-1}\right)\right]=b^{m+1}-a^{m+1} .\right)$

Carson Merrill
Carson Merrill
Numerade Educator
03:46

Problem 41

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.]

Shafiq Rehman
Shafiq Rehman
Numerade Educator
02:46

Problem 42

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 $\phi \circ f$.)

Nick Johnson
Nick Johnson
Numerade Educator
03:29

Problem 43

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.)

Nick Johnson
Nick Johnson
Numerade Educator
01:15

Problem 44

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

Raj Bala
Raj Bala
Numerade Educator
01:04

Problem 45

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 $I_{0}=a .$ )

Carson Merrill
Carson Merrill
Numerade Educator
06:14

Problem 46

(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 $c \in(a, b) .]$

ES
Esraa Samir
Numerade Educator
04:19

Problem 47

(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.)

Uma Kumari
Uma Kumari
Numerade Educator
02:59

Problem 48

(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
$$
\widetilde{S}\left(P_{n}, f g\right):=\sum_{i=1}^{k_{n}} f\left(s_{n, i}\right) g\left(t_{n, i}\right)\left(x_{n, i}-x_{n, i-1}\right)
$$

Nick Johnson
Nick Johnson
Numerade Educator
04:02

Problem 49

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 $m(g)[f(b)-f(a)] \leq \sum_{i=1}^{n} G\left(x_{i-1}\right)\left[f\left(x_{i}\right)-f\left(x_{i-1}\right)\right] \leq$
$M(g)[f(b)-f(a)] .)$

Fahad Paryani
Fahad Paryani
Numerade Educator
02:28

Problem 50

(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 .

Uma Kumari
Uma Kumari
Numerade Educator
02:46

Problem 51

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.)

Nick Johnson
Nick Johnson
Numerade Educator
02:46

Problem 52

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 $D$.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 53

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.

Nick Johnson
Nick Johnson
Numerade Educator
02:08

Problem 54

Let $D$ be a bounded subset of $\mathbb{R}$ and $f: D \rightarrow \mathbb{R}$ be a bounded function. If the boundary $\partial D$ of $D$ is of content zero and if the set of discontinuities of $f$ is also of content zero, then show that $f$ is integrable. In particular, if $D$ is of content zero, then show that $f$ is integrable and its Riemann integral is equal to zero. (Compare Remark 6.8.)

Regina Hays
Regina Hays
Numerade Educator
02:08

Problem 55

Let $f:[a, b] \rightarrow \mathbb{R}$ be a bounded function. If the set of discontinuities of $f$ is of content zero, show that $f$ is integrable. Is the converse true? (Hint: Exercise 34 of Chapter 3 and Example 6.16.)

Regina Hays
Regina Hays
Numerade Educator
02:08

Problem 56

Let $D$ be a bounded subset of $\mathbb{R}$. Let $1_{D}: D \rightarrow \mathbb{R}$ be defined by $1_{D}(x):=1$ for all $x \in D$. Prove the following statements:
(i) $1_{D}$ is integrable if and only if $\partial D$ is of content zero. [Note: If $1_{D}$ is integrable, then $\int_{D} 1_{D}(x) d x$ is called the length of the set $D .]$
(ii) The length of $D$ is zero if and only if $D$ is of content zero.
(iii) If $f: D \rightarrow \mathbb{R}$ is a bounded function and $D_{0} \subseteq D$ is such that $\partial D$ is of content zero, then $f$ is integrable on $D_{0}$.

Regina Hays
Regina Hays
Numerade Educator
02:08

Problem 57

Let $f:[a, b] \rightarrow \mathbb{R}$ be integrable, and $g:[a, b] \rightarrow \mathbb{R}$ 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 6.12.)

Regina Hays
Regina Hays
Numerade Educator