We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Introduction to Analysis : Riemann Integral, Riemann Sums, & Improper Riemann Integration

Study concepts, example questions & explanations for Introduction to Analysis

Create An Account

All Introduction to Analysis Resources

12 Practice Tests Question of the Day Flashcards Learn by Concept

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:

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

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

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

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 of on [a,b] is

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

2. The lower integral of on [a,b] is

$(L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ where is 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.

Report an Error

Example Question #2 : 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:

Partition

Norm

Refinement of a partition

Lower Riemann sum

Upper Riemann sum

Correct answer:

Partition

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 .

Therefore, the term that describes this statement is partition.

Report an Error

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:

Refinement of a partition

Norm

Lower Riemann sum

Partition

Upper Riemann Sum

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,

The norm of 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.

Report an Error

View Tutors

George
Certified Tutor

Pensacola State College, Bachelor in Arts, Mathematics.

View Tutors

Krittika
Certified Tutor

University of Dhaka, Master's/Graduate, Statistics. University of Waterloo, Master's/Graduate, Biostatistics.

View Tutors

Jordan
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Physics. University of Maryland-College Park, Bachelor in Arts, Ger...

All Introduction to Analysis Resources

12 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ISEE Tutors in Chicago, Computer Science Tutors in Denver, ISEE Tutors in Miami, Statistics Tutors in Miami, Algebra Tutors in San Diego, Reading Tutors in Miami, Spanish Tutors in Seattle, SSAT Tutors in New York City, French Tutors in Washington DC, English Tutors in Philadelphia

Popular Courses & Classes

MCAT Courses & Classes in Phoenix, Spanish Courses & Classes in Atlanta, SSAT Courses & Classes in Washington DC, GRE Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Boston, SSAT Courses & Classes in Los Angeles, ACT Courses & Classes in San Diego, Spanish Courses & Classes in New York City, ISEE Courses & Classes in Miami, SSAT Courses & Classes in San Diego

Popular Test Prep

ACT Test Prep in Denver, MCAT Test Prep in Atlanta, LSAT Test Prep in Los Angeles, SSAT Test Prep in New York City, MCAT Test Prep in Boston, SAT Test Prep in New York City, ACT Test Prep in San Diego, GRE Test Prep in New York City, GRE Test Prep in Miami, ISEE Test Prep in San Diego

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Introduction to Analysis : Riemann Integral, Riemann Sums, & Improper Riemann Integration

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 #2 : Riemann Integral, Riemann Sums, & Improper Riemann Integration

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

All Introduction to Analysis Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: