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 – 10 marks

Give a brief account of the relevance of Integration methods to Science and Engineering. At least one paragraph is necessary, stating why and where it is needed. It must be fully referenced.

QUESTION 2 – 20 marks

Explain the method of “Integration By Parts” as simply as possible. Stay away from proofs, think about how you would explain it to a new engineering student. Use your own examples in your explanation to illustrate this method.

QUESTION 3 – 20 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 4 – 20 marks

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

QUESTION 5 – 10 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.

QUESTION 6 – 20 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