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Error Analysis and Simpson's Rule to approximate integrals

  1. Sep 13, 2012 #1
    Hey there,

    This isn't exactly a homework question, but my question did arise while doing homework and I wouldn't know where else to post it, so here I am.

    I'm a Calc 2 student and covered approximations of integrals that I normally wouldn't be able to integrate today. I was going through the homework questions and came across something that took my interest.

    1. The problem statement, all variables and given/known data

    [itex] \int 2x^3 dx [/itex] from 1 to 3 (I don't know how to write the definite integral on this forum)

    [tex] |Error Bound| = \frac{k(b - a)^5}{180n^4} [/tex]

    where a and b are the limits of integration, n is the number of subintervals, and k is the maximum value of the fourth derivative of the integrand.

    Find the error involved in approximating the integral when n = 4 while using Simpson's Rule.

    -

    Now, here's the thing, for this certain function, 2x^3:
    [tex] f'(x) = 6x^2 [/tex]
    [tex] f''(x) = 12x [/tex]
    [tex] f'''(x) = 12 [/tex]
    [tex] f^4(x) = 0 [/tex]

    The fourth derivative is just 0, so the only value k can really take in this case is 0. I thought this was strange, as it meant an approximation with so little partitions would have no error in it, but I checked behind the book through the answers and sure enough, the answer was 0.

    I was still a little puzzled, but I figured that if indeed there's no error in approximating the function, for this case, Simpson's Rule would give the same result as the definite integral. So I went ahead and computed both, and sure enough, the result from both came back as a 40.

    This leaves me with a couple of questions:

    1. Would Simpson's Rule for approximating integrals give the same result as the definite integral in all low-power polynomials such as this?

    2. If that is the case, why is this? Is there a relatively simple reason for this that I'm just not seeing?

    I'm not as familiar with the theory behind this method of approximation so it might be something very simple that I'm just not seeing.

    (Also, yes, I realize there's no point in applying this method since I can just use the Fundamental Theorem of Calculus, but I'm just curious more than anything)


    Thanks for any input
     
    Last edited: Sep 13, 2012
  2. jcsd
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