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Fundamentals of Mathematical Analysis

Rod Haggarty

Chapter 7

Integration - all with Video Answers

Educators

Section 1

The Riemann integral

01:57

Problem 1

Find primitives of the following functions:
(a) $f(x)=x^{2}+3 x-2$
(b) $f(x)=1+\cos 3 x$
(c) $f(x)=\mathrm{c}^{x} \cosh 2 x$
(d) $f(x)=\frac{1}{\sqrt{9-x^{2}}}$
(c) $f(x)={ }_{1} x$

Thomas Emment
Thomas Emment
Numerade Educator
01:09

Problem 2

Let $f(x)=x^{2}$ and $P_{n}$ be the partition $\{0,1 / n, 2 / n, \ldots, 1\}$ of $[0,1]$. Calculate $L\left(P_{n}\right)$ and $U\left(P_{n}\right)$. Hence prove that $f$ is Riemann integrablc on $[0,1]$.

Carson Merrill
Carson Merrill
Numerade Educator
03:55

Problem 3

(iiven that $f(x)=1 / x$ is Riemann integrable on $[1,2]$, calculate $L\left(P_{n}\right)$ and $U\left(P_{n}\right)$, wherc
$$
P_{n}=\left\{1,1+\frac{1}{n}, 1+\frac{2}{n}, \ldots, 2\right\}
$$
Deduce that
$$
\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\frac{1}{2 n} \rightarrow \log _{e} 2 \quad \text { as } n \rightarrow \infty
$$

Taylor Shimono
Taylor Shimono
Numerade Educator
01:02

Problem 4

Prove Propertics (3) and (4) of 7.1.10.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:07

Problem 5

Let
$$
J_{n}=\frac{1}{n} \int_{0}^{1} \frac{\sin n x}{1+x^{2}} \mathrm{~d} x
$$
Use the properties of the Riemann integral to show that $\left|J_{n}\right| \leqslant$ $\pi / 4 n$. Hence evaluate $\lim _{n \rightarrow \infty} J_{n}$

Carson Merrill
Carson Merrill
Numerade Educator
01:46

Problem 6

What is wrong with the following argument? By the fundamental theorem of calculus,
$$
\int_{0}^{2} \frac{1}{(x-1)^{2}} \mathrm{~d} x=\left[\frac{1}{1-x}\right]_{0}^{2}=-2
$$

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
02:52

Problem 7

If $f(x) \geqslant g(x)$ on the interval $[a, b]$, show that the area between the graphs of $f$ and $g$ and the ordinates $x=a$ and $x=b$ is given by $\int_{a}^{b}[f(x)-g(x)] \mathrm{d} x .$ Hence calculate the areas shown in Figure 7.5.

Samriddhi Singh
Samriddhi Singh
Numerade Educator
02:58

Problem 8

Let $f$ be continuous on $[0,1]$ and
$$
\begin{aligned}
&J_{1}=\int_{0}^{1 / \sqrt{\pi}} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x \\
&J_{2}=\int_{1 / \sqrt{n}} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x
\end{aligned}
$$
Use 7.1.12 to prove that
$$
\begin{aligned}
&J_{1}=f\left(c_{n}\right) \tan ^{-1} \sqrt{n} \\
&J_{2}=f\left(d_{n}\right)\left(\tan ^{-1} n-\tan ^{-1} \sqrt{n}\right)
\end{aligned}
$$
where $0<c_{n}<1 / \sqrt{n}<d_{n}<1$. Deduce that
$$
\int_{0}^{1} \frac{n f(x)}{1+n^{2} x^{2}} \mathrm{~d} x \rightarrow \frac{1}{2} \pi f(0) \quad \text { as } n \rightarrow x
$$

Priyanka Sadarangani
Priyanka Sadarangani
Numerade Educator