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# Integral over a subdivided interval Watch

1. Divide [0,1] into N subintervals using .

Let and let p,q be constant functions over [0,1].

I'm trying to find the piecewise solution to the integral . The solution is given in my notes without any explanation so I must be missing something because I'm not sure how to do it. Can someone help?
2. What do you mean by the '+' sign in the definition of phi_k. If it means that (which is my best guess), then for any particular i, there are not many values of j s.t. (and same for the derivatives). It's all fairly straightforward (if a little fiddly).
3. (Original post by DFranklin)
What do you mean by the '+' sign in the definition of phi_k. If it means that (which is my best guess),
That's correct.

then for any particular i, there are not many values of j s.t. (and same for the derivatives).
Can you explain a bit further how you know this? I've just started reading about a new topic today so I may be unfamiliar with this kind of work.
4. (Original post by 0-))
That's correct.

Can you explain a bit further how you know this? I've just started reading about a new topic today so I may be unfamiliar with this kind of work.
Draw a sketch - each function is only non-zero over a small region. For the product to be non-zero the "non-zero" bits of each function will have to overlap.
5. (Original post by DFranklin)
Draw a sketch - each function is only non-zero over a small region. For the product to be non-zero the "non-zero" bits of each function will have to overlap.
OK I can see that the product will be non-zero only if |i-j|=1 or i=j. I'm not how to do the next part - the integration.

Could you show me how to find e.g. . It should all click into place after that.
6. Can anyone help? I'm looking at this for the second time and I'm still struggling.
7. Well, suppose i = j. What is ? There's an interval where it's (which you can make a little nicer by multiplying through by N), and another interval where it's .

Neither of these is terribly difficult to integrate, and then you just add.

is somewhat similar (but I think there's only one interval to consider - I haven't drawn the sketch).
8. (Original post by DFranklin)
Well, suppose i = j. What is ? There's an interval where it's (which you can make a little nicer by multiplying through by N), and another interval where it's .

Neither of these is terribly difficult to integrate, and then you just add.

is somewhat similar (but I think there's only one interval to consider - I haven't drawn the sketch).
I have the solution to for as but after putting the sum of the integrals into Maple I get . Can you see what's going wrong?
9. You've not exactly posted much of your working. But almost certainly, you've got the limits of your integrals wrong.
10. (Original post by DFranklin)
You've not exactly posted much of your working. But almost certainly, you've got the limits of your integrals wrong.
Finally I realise where I've been going wrong. Sorry it took so long!

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