Science:Math Exam Resources/Courses/MATH101/April 2008/Question 02 (b)
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Question 02 (b) 

FullSolution Problem. Justify your answers and show all your work. Simplification of answers is not required. Let be the unbounded region that lies under the curve , above the axis, and to the right of the vertical line . For what values of the constant does the solid obtained by rotating about the axis have finite volume? 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

For a function , the volume of revolution about the xaxis from a to b of this function can be computed by

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let . Then, we wish to know when is
a finite number. To solve this, we compute directly that
This integral has to be treated in two cases. One when and one when . When we have
and this diverges. When , we have
Now, the above limit converges when and diverges when (this is because the term will appear in the denominator in the first case and in the numerator in this second case). Thus, we get convergence only when as required. 