QUESTION 1:
A remarkable fact: The shaded area from x = 1 to infinity is infinite, where revolution is finite. Prove this fact using integration.
QUESTION 2:
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:
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 calculation and you explain them.
QUESTION 4:
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