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Introduction to Analysis : Integrability of Real Numbers (R)

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Example Questions

Example Question #1 :Riemann Integral, Riemann Sums, & Improper Riemann Integration

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

Possible Answers:

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{Z}$

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{Z}$ be bounded

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

, $b\ \epsilon\ \mathbb{R}$ , a>b , and $f:[a,b]\rightarrow \mathbb{R}$

, $b\ \epsilon\ \mathbb{Z}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

Correct answer:

, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded

Explanation:

Using the definition for Riemann sums to define the upper and lower integrals of a function answers the question.

According the the Riemann sum where U(f,P) represents the upper integral and L(f,P) the following are defined:

1. The upper integral ofon [a,b] is

$(U)\int_a^bf(x)dx:=\text{inf}\begin{Bmatrix} U(f,p)) \end{Bmatrix}$ whereis a partition of [a,b] .

2. The lower integral ofon [a,b] is

$(L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ whereis a partition of [a,b] .

3. If 1 and 2 are the same then the integral is said to be

$\int_a^bf(x):=(U)\int_a^bf(x)dx=(L)\int_a^bf(x)dx$

if and only if, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if, $b\ \epsilon\ \mathbb{R}$ , a<b , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

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Example Question #1 :Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

, $b\ \epsilon\ \mathbb{R}$ and a<b . Over the interval [a,b] is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b

Possible Answers:

Lower Riemann sum

Upper Riemann sum

Partition

Norm

Refinement of a partition

Correct answer:

Partition

Explanation:

By definition

If, $b\ \epsilon\ \mathbb{R}$ and a<b .

Apartitionover the interval [a,b] is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b .

Therefore, the term that describes this statement is partition.

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Example Question #3 :Riemann Integral, Riemann Sums, & Improper Riemann Integration

What term has the following definition.

The__________of a partition $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Possible Answers:

Norm

Upper Riemann Sum

Lower Riemann sum

Partition

Refinement of a partition

Correct answer:

Norm

Explanation:

By definition

If, $b\ \epsilon\ \mathbb{R}$ and a<b .

A partition over the interval [a,b] is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

a=x_0<x_1<...<x_n=b .

Furthermore,

Thenormof the partition

$P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Therefore, the term that describes this statement is norm.

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Introduction to Analysis : Integrability of Real Numbers (R)

Study concepts, example questions & explanations for Introduction to Analysis

All Introduction to Analysis Resources

Example Questions

Example Question #1 :Riemann Integral, Riemann Sums, & Improper Riemann Integration

Example Question #1 :Riemann Integral, Riemann Sums, & Improper Riemann Integration

Example Question #3 :Riemann Integral, Riemann Sums, & Improper Riemann Integration

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