QUESTION 1 – 25 marks
A remarkable fact: The shaded area from x = 1 to infinity is infinite, wh
revolution is finite. Prove this fact using integration.
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 meth
How to calculate areas and volumes
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