Math Is Fun Forum

Au101

Member

Registered: 2010-12-01

Posts: 353

Riemann Sums

Hi guys, I have a problem with understanding Riemann sums. I have read the wikipedia article on the subject and I can essentially understand what they are, my problem is with understanding a particular integral. The derivation which wikipedia gives requires a knowledge of Riemann sums which I don't seem to have and I would be most grateful if somebody could explain the derivation to me.

I have been working on polar coordinates and have reached the section on the area of a sector, which is given by the integral:

And the derivation of this integral is given by wiki as follows:

This I can understand, but the final part of the derivation I do not follow:

Is there anybody who could help to explain this last step? Thanks

Last edited by Au101 (2011-08-13 03:06:07)

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Riemann Sums

Sorry about the strange line breaks and my failure to get the LaTeX up originally, I had left in a hyphen as the minus instead of a a dash. That's the trouble with copying plain text in

zetafunc.

Guest

Re: Riemann Sums

Sorry, this is probably wrong, but isn't it just because if you have some function and take the area of an infinite number of infinitesimally small sub-intervals you will have calculated the Riemann sum for the integral? Sort of like taking a curve and drawing more and more rectangles until you have drawn so many infinitesimally small rectangles that the total area left is 0, which would be the same in polar co-ords too? That would be the Riemann sum, would it not?

Sorry if this sounds stupid or you already knew this...

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Riemann Sums

Nono I think you're right, I'm just having difficulty explaining the jump from:

It seems to make intuitive sense in my head, that as n tends to infinity that is the integral which you will end up with. The thing is though, at my school you could do double or single maths. I did single maths which means I didn't cover what is known as further pure, which has a lot of interesting pure maths on it which I wanted to cover. So I got a textbook and have spent my recent holidays on the first two of three modules. My method has been to write myself my own 'sort-of textbook', which is more a sort of extended set of notes with various tangents and exercises and what I wanted to do was to explain this derivation. The thing is, it makes some sense in my head, but I certainly can't write-up an explanation without sort of waving my hands and saying, well it looks about right.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Riemann Sums

Think of the area of a circle and the definition of pi.

Rieman sum is defined on adding up the infinite amount of infinitesimals, which comes from finite amount of finite divisions. The finite sum varies as the division goes finer, but it seems to approach a certain quantity, which is just defined as the Rieman integration.

This is similar to define the area of a circle, or the circle itself.

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X'(y-Xβ)=0

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Riemann Sums

I see, okay, I think I have it figured out, now, thank you

SolutionSeeker

Member

Registered: 2011-11-17

Posts: 1

Re: Riemann Sums

I need help solving the following story problem?

Two trains 150 miles apart are traveling toward each other along the same track. The first train goes 60 miles per hour; the second train rushes along at 90 miles per hour. A fly is hovering just above the nose of the first train. It buzzes from the first train to the second train, turns around immediately (assume zero turnaround time), flies back to the first train, and turns around again. It goes on flying back and forth between the two trains until they collide. If the fly’s speed is 120 miles per hour, how far will it travel (answer is in miles).

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Riemann Sums

HI;

Look at it like this:

It will take the two trains 1 hour to collide. The fly's speed is 120 mph. It will be flying
for 1 hour. It travels 120 miles.

Welcome to the forum.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.