Is Simpson's Rule Effective for Solving the Intersecting Cylinders Problem?

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Best numerical technique?

I've recently used Simpson's (1/3) rule for the numerical solution of the 'intersecting cylinders' problem. I've found that it isn't too accurate no matter how many intervals I take (I have even taken 1000,000 intervals!), but still end up with some error.

I'll appreciate anyone for helping me in this matter
 
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What is the intersecting cylinders problem?
 
it is discussed in the thread called 'intersecting cylinders' in this forum. the results of numerical integration do converge but always with some error
 
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