Analysis-II

Teacher

Dr. Bhairaba Kumar Majhi

Category

Core Courses

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Course Name : Analysis-II

Code(Credit) : CUTM-1515(3-0-3)

Course Objectives

  • To introduce Riemann integrable and Riemann sums

  • To describe various theorems about Riemann sums and Riemann integrals and emphasize the proofs’ development.

  • To evaluate the definite integral, double integral and triple integral.

Learning Outcomes

  • Upon successful completion of this course, students will be able to:

  • DefineIntegrability of continuous,Piecewise Continuous and Monotone Functions.

  • Be competent in solving Improper Integrals, Double integration and triple integrals over different region.

  • Applychange of variables in double integrals and triple integrals.

Course Syllabus

Module-I 

Improper Integrals Definitions and Types; Convergence of Unbounded Functions with Finite Limit of Integration; Comparison Test For Convergence and Beta Function; General Test for Convergence and Absolute Convergence For Finite Range Integrals; Examining Convergence of Finite Range Integrals.

Project 1: A Report on Beta Function,its Properties and Application.

Project 2: A Report on Different types of Comparison Tests in Improper Integrals

Project 3: A Report on Convergence of Finite Range Integrals.

Module-II 

Infinite Range  Integrals and Convergence of Infinite Range of Integrals; Comparison Test For Convergence and Gamma Function; General Test for Convergence and Absolute Convergence For Infinite Range; Problems on Examining Convergence of Infinite Range Integrals.

Project 4: A Report on Gamma Function, its Properties and Application.

Project 5: A Report on Convergence of Infinite Range Integrals.

Module-III 

Double Integrals Definition and Techniques of Evaluation; Double Integration Over a Rectangle; Condition Of Integrability and Some Theorems; Calculation Of Double Integral Over a Rectangle.

Project 6: A Report on Double Integrals with its Properties and Evaluation Techniques.

Project 7: A Report on Setting  up Limits of Integration for Evaluating double Integrals

Project 8: A Report  on Area calculated as A Double Integral and Finding Area of Regions.

Module-IV 

Double Integral Over a Region(Closed Bounded Domain); Calculation Of Double Integral Over a Closed Domain; Calculation Of Double Integral by Changing the Order Of Integration; Change of Variable in Double Integrals; Computing Double Integrals by Change of variable Techniques.

Project   9: Test of Convergence of Improper Integral for Finite range of Integration

Project 10: Test of Convergence of Improper Integral for Infinite range of Integration

Project 11:  A collection of Different Test Criteria for Convergence of Improper Integrals

Module-V

Double Integrals in Polar Co-ordinates; Computing Double Integrals in Polar Co-ordinates ;Triple Integrals Concepts and Definitions ;Triple Integrals over a Parallelepiped; Calculation of a Triple integrals Over a Parallelepiped.

Project  12: A Report on Application of Double Integrals in Polar Co-ordinates.

Project  13 :A Reports on Triple Integrals and its Applications to Solve Problems.

Project  14: A Report on Some Important Theorems on Triple Integrals.

Module-VI 

Triple Integral Over Regions(Bounded Domaines); Volume Of Solids by Triple Integrals; Calculation of a Triple Integrals Over any Region on R3.

Project 15:  A Report on Solving  Problems on Triple integrals over a Parallelepiped .

Project 16:  A Study on Volume Calculated as A triple Integral and Calculating Volume Of Solids.

Module-VII 

Triple Integrals in Cylindrical and Spherical Co-ordinates; Problems on Cylindrical and Spherical Co-ordinates; Change of variable in Triple integrals ; Computing Triple Integrals by Change of variable Techniques.

Project 17: A Report on Determining Limits of Integration for Cylindrical Co-ordinates in triple Integrals.

Project 18: A Report on  Change of variable in Triple Integrals with the help of Jacobians.

 

Text Book:

C. Mallik& S. Arora,Mathematical Anyalysis, New Age Pub. House New Delhi. Chapters: 11, 17 (2.1 to 2.8, 3.1 to 3.4, 5.1 to 5.2),  18 (7.1 to 7.7)

 

Reference Books:

Calculus Early Transcendentals 10th Ed - Howard Anton, Iril Bivens & Stephen Davis .Joh Wiley & Sons.

Chapter : 14 (Except 14.4 and 14.8).

Topics In Calculus By R.K Panda & P.K Satapathy, S.G Publication,Puri

 

Session Plan

Session 1:

Session 2:

Convergence of Unbounded Functions with Finite Limit of Integration.

https://www.youtube.com/watch?v=DkWSsckVrpg

Session 3:

Session 4:

General Test for Convergence and Absolute Convergence For Finite Range.

https://www.youtube.com/watch?v=rEyeRKd3TWo&t=4s

Session 5:

Examining Convergence of Finite Range Integrals.

Session -(6 & 7):

Project 1:

A Report on Beta Function,its Properties and Application.

https://www.youtube.com/watch?v=yyo798JxN3c

https://www.youtube.com/watch?v=KZuhYiNQbpQ

Session -(8 & 9):

Project 2:

A Report on Different types of Comparison Tests in Improper Integrals

https://www.youtube.com/watch?v=-DHgr52BHlc

Session -(10 & 11):

Project 3:

A Report on Convergence of Finite Range Integrals.

Session 12:

Infinite Ramge Integrals and Convergence of Infinite Range of Integrals

https://www.youtube.com/watch?v=7VhyzlDzntQ

Session 13:

Session 14:

General Test for Convergence and Absolute Convergence For Infinite Range.

Session 15:

Problems on Examining Convergence of Infinite Range Integrals.

https://www.youtube.com/watch?v=VRCNnIJ2cWU

Session -(16 & 17):

Project 4:

A Report on Gamma Function, its Properties and Application.

https://www.youtube.com/watch?v=PwCl7vVyXwY&t=38s

Session -(18 & 19):

Project 5:

A Report on Convergence of Infinite Range Integrals.

Session 20:

Double Integrals Definition and Techniques of Evaluation

https://www.youtube.com/watch?v=UubU3U2C8WM&t=309s

Session 21:

Double Integration Over a Rectangle

https://www.youtube.com/watch?v=6qSt6Gu6R5M

Session 22:

Session 23:

Calculation Of Double Integral Over a Rectangle

https://www.youtube.com/watch?v=QY7WFlTa8_U

Session -(24 & 25):

Project 6:

A Report on Double Integrals ,its Properties and Evaluation Techniques.

 

Session -(26 & 27):

Project 7:

A Report on Setting  up Limits of Integration for Evaluating double Integrals

https://www.youtube.com/watch?v=K27g2mqjIao

Session -(28 & 29):

Project 8:

A Report  on Area Calculated as A Double Integral and Finding Area of Regions,

https://www.youtube.com/watch?v=xec6HTcn2M8

https://www.youtube.com/watch?v=nwFBRmzmFJk

Session 30:

 

Session 31:

Calculation Of Double Integral Over a Closed Domain

https://www.youtube.com/watch?v=9xUyOINEQio

Session 32:

Calculation Of Double Integral by Changing the Order Of Integration

https://www.youtube.com/watch?v=K27g2mqjIao&t=44s

Session 33:

Change of Variable in Double Integrals

Session 34:

Computing Double Integrals by Change of variable Techniques

Session -(35 & 36):

Project 9:

Test of Convergence of Improper Integral for Finite Range of Integration

Session -(37 & 38):

Project 10:

Test of convergence of improper integral for infinite range of integration

Session -(39 & 40):

Project 11:

A collection of Different Test criteria for convergence of Improper Integrals

Session 41:

Double Integrals in Polar Co-ordinates

https://www.youtube.com/watch?v=YQfta_poP1k

 

Session 42:

Computing Double Integrals in Polar Co-ordinates

Session 43:

Triple Integrals Concepts and Definitions

https://www.youtube.com/watch?v=CPR0ZD0IYVE

Session 44:

Triple Integrals over a Parallelepiped.

Session 45:

Calculation of a Triple integrals Over a Parallelepiped.

Session -(46 & 47):

Project 12:

A Report on Application of Double Integrals in Polar Co-ordinates.

https://www.youtube.com/watch?v=a30gTO1q3Aw

Session -(48 & 49):

Project 13:

A Reports on Triple Integrals and its Applications to Solve Problems.

https://www.youtube.com/watch?v=1I9IcDD-Hc4

Session -(50 & 51):

Project 14:

A Report on Some Important Theorems on Triple Integrals.

 

Session 52:

Triple Integral Over Regions(Bounded Domaines)

https://www.youtube.com/watch?v=ly35U621we8

 

Session 53:

Volume Of Solids by Triple Integrals

https://www.youtube.com/watch?v=AQHcZklcltI

Session 54:

Calculation of a Triple Integrals Over any region on R3.

https://www.youtube.com/watch?v=hITDhcLbae0

 

Session -(55 & 56):

Project 15:

A Report on Solving  Problems on Triple integrals over a Parallelepiped .

https://www.youtube.com/watch?v=XT7jKBZyjHc

Session -(57 & 58):

Project 16:

A Study on Volume Calculated as A triple Integral and Calculating Volume Of Solids.

https://www.youtube.com/watch?v=-r6LuD80iKI

https://www.youtube.com/watch?v=tsbFhR-A_JA

Session 59:

Triple Integrals in Cylindrical and Spherical Co-ordinates

https://www.youtube.com/watch?v=UTIjALWr6GM

 

Session 60:

Session 61:

Change of variable in Triple integrals

https://www.youtube.com/watch?v=xHE-WI6z3Uk

Session 62:

Computing Triple Integrals by Change of variable Techniques.

https://www.youtube.com/watch?v=xHE-WI6z3Uk&t=50s

Session -(63& 64):

Project 17:

A Report on Determining Limits of Integration for Cylindrical Co-ordinates in triple Integrals.

Session -(65 & 66):

Project 18:

A Report on  Change of variable in Triple Integrals with the help of Jacobians.

https://www.youtube.com/watch?v=5WGW82fSXnU

 

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    Our Main Teachers

    My Self Dr.Bhairaba Kumar Majhi,M,Sc,B.Ed,M.Tech(CSE),Ph.D(Mathematics ) ,Asst.Professor And HOD Department of Mathematics School Of Applied Sciences,Presently working at CUTM,Bolangir Campus.I have 16 Years of Experience in Teaching at Different Levels Of UG,PG & Engineering Mathematics.