MATH200 December 2011
• Q1 (a) • Q1 (b) • Q1 (c) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 • Q5 (a) • Q5 (b) • Q6 • Q7 • Q8 (a) • Q8 (b) i • Q8 (b) ii • Q8 (b) iii •
Evaluate the triple integral
where E is the region in the first octant bounded by the parabolic cylinder and the planes .
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!
The hardest part here is to find the boundaries for the region of integration. Often it is very helpful to make a sketch, like this (boundaries of the region are in red):
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.
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The plane intesects at the line . Since the region is is the first octant, we know that .
Using this information and the sketch in the hint, we find the region as
Hence the integral is
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