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
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To introduce Riemann integrable and Riemann sums
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To describe various theorems about Riemann sums and Riemann integrals and emphasize the proofs’ development.
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To evaluate the definite integral, double integral and triple integral.
Course Outcomes
| COs | Course outcomes | Mapping Cos with POs (High-3, Medium-2, Low-1) |
| CO1 | Define Integrability of continuous, Piecewise Continuous and Monotone Functions.
|
PO1(3), PO2(2) |
| CO2 | Be competent in solving Improper Integrals, Double integration and triple integrals over different region.
|
PO2(2), PO4(2) |
| CO4 | Apply change of variables in double integrals and triple integrals
|
PO1(2), PO4(2) |
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:
Improper Integrals Definitions and Types.
Session 2:
Convergence of Unbounded Functions with Finite Limit of Integration.
Session 3:
Comparison Test For Convergence and Beta Function.
Session 4:
General Test for Convergence and Absolute Convergence For Finite Range.
Session 5:
Examining Convergence of Finite Range Integrals.
Session -(6 & 7):
Project 1:
A Report on Beta Function,its Properties and Application.
Session -(8 & 9):
Project 2:
A Report on Different types of Comparison Tests in Improper Integrals
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
Session 13:
Comparison Test For Convergence and Gamma Function.
Session 14:
General Test for Convergence and Absolute Convergence For Infinite Range.
Session 15:
Problems on Examining Convergence of Infinite Range Integrals.
Session -(16 & 17):
Project 4:
A Report on Gamma Function, its Properties and Application.
Session -(18 & 19):
Project 5:
A Report on Convergence of Infinite Range Integrals.
Session 20:
Double Integrals Definition and Techniques of Evaluation
Session 22:
Condition Of Integrability and Some Theorems
Session 23:
Calculation Of Double Integral Over a Rectangle
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
Session -(28 & 29):
Project 8:
A Report on Area Calculated as A Double Integral and Finding Area of Regions,
Session 30:
Double Integral Over a Region(Closed Bounded Domain)
Session 31:
Calculation Of Double Integral Over a Closed Domain
Session 32:
Calculation Of Double Integral by Changing the Order Of Integration
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 42:
Computing Double Integrals in Polar Co-ordinates
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.
Session -(48 & 49):
Project 13:
A Reports on Triple Integrals and its Applications to Solve Problems.
Session -(50 & 51):
Project 14:
A Report on Some Important Theorems on Triple Integrals.
Session 52:
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 .
Session -(57 & 58):
Project 16:
A Study on Volume Calculated as A triple Integral and Calculating Volume Of Solids.
Session 59:
Triple Integrals in Cylindrical and Spherical Co-ordinates
Session 60:
Problems on Cylindrical and Spherical Co-ordinates
Session 62:
Computing Triple Integrals by Change of variable Techniques.
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.
Case Studies
Case Studies
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.