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Is everything integradable?

Reply 20

shiny

12

elpaw

im not scared, just fr :eek:

ed out

. nah, it just looks so complicated to someone who has only just met PDEs

My PhD is probably going to be about solving PDEs numerically :frown:

Reply 21

mathz

8

mockel

hmmm, how does the fact that f(x)=1 if x is rational and f(x)=0 if x is irrational prove significant in this question?

tough question.
to define Riemann integration under curve on [a,b] you use idea of upper and lower sums. take a curve btween a and b and draw rectangles of a set width to the curve.some will be under the curve some over (think trap rule or simpsons rule) .we can define in a way for rectangles to be above the curve (for any width take 1 corner of the rectangle to be max value of curve) and define rectangles all to be below the curve (take min value of curve).the integral is then defined if the sum of the area of the rectangles above the curve (upper sum) tends to the sum of the area of the rectangles below the curve(lower sum) as the width of the rectangles tend to 0 (not very mathematical ,not good to consider 2 sequences tending to each other but clearer to see what is going on.should have said upper sum-lower sum tends to 0). this area is then defined to be the integral between a and b of the the curve.
for the function in above post any rectangle for the upper sum will have a rational in it so area will be 1.width of rectangle since for interval size 1/n on [0,1] will have n rectanlges so upper sum =n.1.1/n=1.any rectangle for the lower sum will have an irrational in it so the area will be 0 an lower sum is 0. therefore the upper sum will not tend to the lower sum so function not Riemann integrable.

Reply 22

m:)ckel

14

mathz

for the function above any rectangle for the upper sum will have a rational in it so area will be 1.any rectangle for the lower sum will have an irrational in it so the area will be 0. therefore the upper sum will not tend to the lower sum so function not Riemann integrable.

i don't really understand this bit :redface:

Reply 23

JamesF

1

mockel

i don't really understand this bit :redface:

The Riemann integral only exists if the upper and lower sums, converge to the same limit.

Basically, what you are doing is defining a sum which is larger than the integral and a sum which is smaller than the integral, then the value of the integral itself must be somewhere in between these values, and so if these both tend to the same limit, the integral must be this value. So if they dont tend to the same limit, the Riemann integral doesnt exist.

Reply 24

RobbieC

15

shiny

Why you so scared of Complex Analysis? You Physicists love that sort of thing :smile:

I dont. Standard integration makes me cry :frown:

Reply 25

m:)ckel

14

JamesF

The Riemann integral only exists if the upper and lower sums, converge to the same limit.

Basically, what you are doing is defining a sum which is larger than the integral and a sum which is smaller than the integral, then the value of the integral itself must be somewhere in between these values, and so if these both tend to the same limit, the integral must be this value. So if they dont tend to the same limit, the Riemann integral doesnt exist.

okay, i get that- but how does this link in with the rational and irrational things mentioned earlier?

Reply 26

JamesF

1

mockel

okay, i get that- but how does this link in with the rational and irrational things mentioned earlier?

If the limit of one sum is rational, and the other is irrational, then they cannot be equal, so the function is not Riemann integrable.

Reply 27

Jump

18

RobbieC

I dont. Standard integration makes me cry :frown:

Ditto!

Reply 28

m:)ckel

14

JamesF

If the limit of one sum is rational, and the other is irrational, then they cannot be equal, so the function is not Riemann integrable.

ah right- it's starting to make sense now :smile:

later on in the pure a-level syllabus, you learn numerical methods for solving integrals- are these still riemann integrable?

Reply 29

JamesF

1

JamesF

If the limit of one sum is rational, and the other is irrational, then they cannot be equal, so the function is not Riemann integrable.

I dont actually know how mathz has got that the upper sum will be rational and the lower will be irrational :confused:

, i was just explaining the rest lol.

Reply 30

mathz

8

mockel

ah right- it's starting to make sense now :smile:

later on in the pure a-level syllabus, you learn numerical methods for solving integrals- are these still riemann integrable?

sorry for confusion ,i need more coffee.have corrected my previous post.its easier to see this "strange" way of defining integration with pictures.

Reply 31

m:)ckel

14

thanks, it's starting to make sense :smile:

i'm sure i'll fully understand it in time

Reply 32

terryb

6

There are far more functions that cannot be integrated than those that can be. The reason that everything looks so rosy at school (and even lower level University) is that, to make a question worthwhile, it has to be integrable. :rolleyes:

Incidentally, everything so far has been about indefinite integrals. (These are ones which don't have limits.) It's a whole new ball game dealing with definite integrals, and the number of integrals that cannot be evaluated drops considerably. Many of these methods can be solved very quickly using a computer package.

Is everything integradable?

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