Proving Positive Integral using Sums

  1. 1. The problem statement, all variables and given/known data
    Prove that the integral from 0 to x of (sin t)/(t + 1) dt > 0.
    (Sorry I don't know how to use the integral latex.)

    2. Relevant equations
    We have only learned about lower and upper sums, and how the integral is equal to the supremum of lower sums and the infimum of upper sums when sup = inf

    3. The attempt at a solution
    I took a partition from 0 to x with equal intervals, so that each interval has length x/n for n intervals. I found the lower sum and upper sum, with a difference of x/n * (sin x)/(x + 1). I have no idea what to do afterwards. Am I to eliminate the possibility of the integral being 0 and negative?

    EDIT: I "found" the lower and upper sums incorrectly - in fact, they are not possible to determine.
    Last edited: Jan 7, 2010
  2. jcsd
  3. Mark44

    Staff: Mentor

    Your integral is a function of x, f(x), with f(0) = 0. If you can show that f'(x) > 0 then you can conclude that f is increasing, hence its integral will be positive.

    Here's the LaTeX for your integral.
    [tex]\int_0^x \frac{sin t}{t + 1}dt[/tex]
  4. Wouldn't that exclude the decreasing portion of the function for x > 0, since sine is periodic? Moreover, it wouldn't show that the difference between the area of the first and second arcs is positive.
  5. Mark44

    Staff: Mentor

    The function I'm talking about is not (sin t)/(t + 1), which does oscillate as you describe. The one I'm talking about is the integral. The area under the first arch is positive, and is more positive than the negative "area" above the second arch, so overall for the two arches, the integral is positive. It's sort of like adding 5 - 2.5 + 1.25 - .625 + - ... and so on. Each partial sum is positive, despite half the terms being negative.

    I believe the approach you need to take is as I described in my previous post.
  6. Ohhhh. I will try that then, thanks.

    I got
    lim as h -> 0 of [tex]\frac{f(x + h) - f(x)}{h}[/tex] =
    lim as h -> 0 of [tex]\frac{\int_x^(x+h) \frac{sin t}{t + 1}dt}{h}[/tex]

    I'm stuck here, as I tried to equate the limit to a constant c greater than zero, then using the delta-epsilon definition of the derivative. Then I get a delta dependent on the value of x, which means that the function is not uniformly continuous (but it must be because it is continuous in a closed interval).
    Last edited: Jan 7, 2010
  7. Dick

    Dick 25,914
    Science Advisor
    Homework Helper

    The integral of sin(t)/(1+t) is positive from 0 to pi. No doubt about that, right? Now the integral from pi to 2pi is negative. But it will be less in absolute value than the positive part of the integral from 0 to pi. Since 1/(1+t) is less on the interval [pi,2pi] than it is on [0,pi]. Similarly the integral on [2pi,3pi] is positive, integral on [3pi,4pi] is negative but less in absolute value than the former. Etc etc. That's what Mark44 was talking about. It's not really a derivative problem.
  8. But that's not really a proof right? That's just an intuitive solution.
    Also, we have not learned how to differentiate an integral.
  9. Dick

    Dick 25,914
    Science Advisor
    Homework Helper

    It's not just an 'intuitive solution'. It's an informal description of a legitimate proof. How formal you want to make it depends on the requirements of your class. I think if you understand that and describe it clearly, that's all you need.
  10. I was thinking of showing the same description for the proof. Usually, my professor is lenient but the people marking are very harsh and always demand a formal proof.
  11. Dick

    Dick 25,914
    Science Advisor
    Homework Helper

    I think you are being overly concerned. You can't prove the derivative of your integral is positive, because it's not. It does decrease sometimes but can never decrease to zero because of the area argument. Some things aren't worth being overly formal about.
  12. All right then, thanks to the both of you for your help!
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