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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:
Why is integration needed
How to solve integrals using advanced meth
How to calculate areas and volumes

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