# Analysis-II

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

## 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 2:

Convergence of Unbounded Functions with Finite Limit of Integration.

## Session 4:

General Test for Convergence and Absolute Convergence For Finite Range.

## Session 5:

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.

## Session 12:

Infinite Ramge Integrals and Convergence of Infinite Range of Integrals

## Session 14:

General Test for Convergence and Absolute Convergence For Infinite Range.

## Session 15:

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.

## Session 20:

Double Integrals Definition and Techniques of Evaluation

## Session 21:

Double Integration Over a Rectangle

## Session 23:

Calculation Of Double Integral Over a Rectangle

## Project 6:

A Report on Double Integrals ,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,

## 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

## 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

## Session 41:

Double Integrals in Polar Co-ordinates

## Session 42:

Computing Double Integrals in Polar Co-ordinates

## Session 43:

Triple Integrals Concepts and Definitions

## Session 44:

Triple Integrals over a Parallelepiped.

## Session 45:

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.

## Session 52:

Triple Integral Over Regions(Bounded Domaines)

## Session 53:

Volume Of Solids by Triple Integrals

## Session 54:

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.

## Session 59:

Triple Integrals in Cylindrical and Spherical Co-ordinates

## Session 61:

Change of variable in Triple integrals

## Session 62:

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.

## Case Studies

### Dr. Bhairaba Kumar Majhi

##### Asst.Professor
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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.