INTEGRATION ASSIGNMENT

• The presentation of your answers matters a lot – you must explain what you are doing and you must use proper mathematical notation (as used in texts, notes etc). Just writing an answer without working is not enough.

Guidelines for submitted work

• This is an academic assignment so it must be referenced. Include a Reference list at the end of your assignment. You must cite all the different authors from different sources (books, journals, electronic). Please do not use Wikipedia or other such non-refereed sources.

• Start each new question on a new page.

• Conduct a spell check yourself and ensure you have a critical friend read and comment on your English usage, grammar, punctuation and other technical issues.

• Please use Microsoft Word Equation to express any mathematical formulas needed, and Microsoft Word to write the assignment or similar software.

• You may use graphical or drawing software to show your graphs.

• Please use size 12 font for any written work.

• Leave a wide margin on the left for feedback comments from your teachers.

What is plagiarism

Plagiarism is when you copy someone else’s answers. Even if you make slight changes in symbols it is still plagiarism. Plagiarism is cheating and is wrong. If it is detected all the people whose answers are extremely similar will get zero marks for the questions involved.

It is a good idea to discuss problems with other people. It is often helpful to work in study groups, but you must write up your answers by yourself and your examples must be unique. Any similarities identified with TURNITIN will be investigated and penalties will be applied.

Integration Assignment

You have been hired to write an introduction to the section on Integration for the Project

“ Mathematics 1B Text Book ”.

NO CALCULATORS TO BE USED, THAT IS, USE NUMBERS THAT ARE SMALL AND EASY TO CALCULATE MANUALLY.

QUESTION 1 – 25 marks

A remarkable fact: The shaded area from x = 1 to infinity is infinite, whereas its volume of revolution is finite. Prove this fact using integration.

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QUESTION 2 – 25 marks

Explain when is the method of “Completing the Square” useful for integration, with your own examples. Make sure you show the step-by-step calculations and you explain them.

QUESTION 3 – 25 marks

Explain the method of “Partial Fractions” for solving integrals for irreducible quadratic fractions. Use your own examples in your explanation. Make sure you show the step-by-step calculations and you explain them.

QUESTION 4 – 25 marks

Consider the circle, centre (0, a), and a radius of 1 unit. The solid of revolution that will be obtained if the circle is revolved about the x-axis is a Torus (a doughnut to most people). Use integration to find its volume. The example on the picture below shows a circle with a centre at (0, 3). In your calculations do not use 3, use “a”. You may show your calculations with other values for “a”, but not 3. In your conclusion, your answer should be in terms of “a” and not any particular value.

Performance Objectives: Know:

Why is integration needed

How to solve integrals using advanced methods of integration How to calculate areas and volumes