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Definition of an integral question

  1. Feb 25, 2013 #1
    definition of an integral question....

    if f(x) ≥ 0, how can you use the definition of an integral to prove that ∫(a,b)f(x)dx ≥ 0?

    This seems like it is an easy question, and seems like one of those things that seems obvious but hard to explain, and the only definition of an integral Ive been able to find in my text book is for a definite integral that I didn't find useful.
     
  2. jcsd
  3. Feb 25, 2013 #2
    Re: definition of an integral question....

    What was the definition? If you look closely, ∫(a,b)f(x)dx is a definite integral!
     
  4. Feb 25, 2013 #3
    Re: definition of an integral question....

    http://img42.imageshack.us/img42/5464/definitionofintegral.png [Broken]

    was the definition I found in my textbook, which i dont understand all that well, i can evaluate an integral but im not very good an interpreting definitions like this :/
     
    Last edited by a moderator: May 6, 2017
  5. Feb 25, 2013 #4
    Re: definition of an integral question....

    Are you familiar with the Riemann Sum definition of an integral? Try looking at a Riemann sum when f(x) is always > (or equal to) 0.

    Edit: I replied before I saw OP's response, what does f(x) > 0 tell you in the sum?
     
  6. Feb 25, 2013 #5
    Re: definition of an integral question....

    no have not heard of a Riemann Sum, but will look into it :)

    -.- i dont see a response by OP but alright :P
     
  7. Feb 25, 2013 #6
    Re: definition of an integral question....

    OP = original poster = you!

    What Vorde is saying is that you're given that f(x) ≥ 0, and you have the function f in the summation. What can you do from there?
     
  8. Feb 25, 2013 #7

    Zondrina

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    Re: definition of an integral question....

    If you know f(x) ≥ 0 for all x in [a,b], then using Riemann sums it's easy to show that the integral of f(x) over the interval [a,b] must also be ≥ 0.

    Start by partitioning your interval [a,b] into n equal sub-intervals. What can you tell me about ##f(x_{i}^{*})## for any choice of ##x_{i}^{*}## in any sub-interval of your choice ##[x_{i-1}, x_i]##
     
  9. Feb 25, 2013 #8
    Re: definition of an integral question....

    ah k apparently have learned about that, found my notes on riemann sum, just didnt know it by name, from much ealier in the course. gonna see what i can come up with with you guys' help so far and my notes :)
     
  10. Feb 25, 2013 #9
    Re: definition of an integral question....

    If f(x) ≥ 0 for all x in [a,b], then f(xi) ≥ 0 and Ʃf(xi*) ≥ 0.

    and if all values for x between [a,b] and Δx = (b-a)/n then Δx ≥ 0 as well.

    So, Ʃf(xi*))Δx ≥ 0. and if all values are ≥ 0, then the limit must be as well?

    then ∫(a,b)f(x)dx = limn→∞Ʃf(xi*)Δx ≥ 0
     
  11. Feb 25, 2013 #10

    Zondrina

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    Re: definition of an integral question....

    Just a small add on. If f(x) ≥ 0 for all x in [a,b], then ##f(x_{i}^{*}) ≥ 0## for any ##x_{i}^{*}## in any sub-interval you choose for all i.

    So yes, the sum will behave in the same manner.

    Also, Δx is assumed to be positive regardless... it wouldn't make sense for the length of your interval to be negative right?

    So the Riemann sum is always greater or equal to zero and even as n→∞, the sum will still be ≥ 0. Though as n→∞, you get the integral over [a,b] which is also ≥ 0 as a result.
     
  12. Feb 25, 2013 #11
    Re: definition of an integral question....

    http://www.cliffsnotes.com/study_guide/Definite-Integrals.topicArticleId-39909,articleId-39903.html [Broken]

    This page has a pretty good explanation.
    Basically you are taking the sum as n goes to infinity (the number of squares) that are infinitesimally small (dx) of some function (usually f(xi) or f(xi*))
    Which leads to:
    [tex]\lim_{n \to \infty} \sum_{i=1}^{n} f(x_{i*})\Delta x[/tex]

    That's my definition :P
     
    Last edited by a moderator: May 6, 2017
  13. Feb 25, 2013 #12
    Re: definition of an integral question....

    Ah ok, n ya good point :P

    thanks for the help everyone :)
     
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