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HSC 2015 MX2 Integration Marathon (archive) (4 Viewers)

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glittergal96

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Re: MX2 2015 Integration Marathon

A kind of open-ended question slightly related to my second method above:

Explain why Simpson's rule with equal spacing is "better" as a tool to numerically approximate definite integrals of smooth functions (functions with continuous derivatives of all orders) than the trapezoidal rule with equal spacing.

Try to use calculus and rigorous statements to justify your explanation, rather than just saying vague qualitative things like "quadratics can approximate curves better than lines".
 
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seanieg89

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Re: MX2 2015 Integration Marathon

Good stuff glittergal, dyadic decomposition is exactly what I wanted someone to come up with for estimating that integral. (And your exponent is the correct one). :).
 

VBN2470

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Re: MX2 2015 Integration Marathon

^ Use the substitution to simplify your integral to .
 
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glittergal96

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Re: MX2 2015 Integration Marathon

Here is one of the proofs I had in mind for my factorial question:







(Of course, we could take much cruder upper bounds of the Basel sum if we do not want to assume the well known series evaluation. For example we could use an integral to bound that sum too! The main point is that it is finite.)
 

mrpotatoed

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Re: MX2 2015 Integration Marathon

http://integrals.wolfram.com/index.jsp?expr=(x*(sqrt(1-(x-1)^2)&random=false

^ can anyone help with that? Not sure how to sure the latex thingy.

that will show the equation in full, but doesn't include the upper limit of 2 and lower of 0. Any help in working it through appreciated.

nvm... got it.. will leave it up though anyway if anyone wants to try themself
 
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porcupinetree

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Re: MX2 2015 Integration Marathon

Or using area of a quarter of a circle with radius 1
That's one of the most enjoyable moments, using that trick/shortcut.

Here's a cool reduction formula question:

If

Find a reduction formula for and hence, find
 

glittergal96

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Re: MX2 2015 Integration Marathon

That's one of the most enjoyable moments, using that trick/shortcut.

Here's a cool reduction formula question:

If

Find a reduction formula for and hence, find
Too lazy to do the calculation at the end, but the reduction formula for n >= 2:

 
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glittergal96

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Re: MX2 2015 Integration Marathon

Ps, you should use a "\" before commonly used trig functions in latex to make things look nicer.
 
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