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Math Help - Numerical Integration

  1. #1
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    Numerical Integration

    Determine theminimum number of subintervals to estimate the value of integral(8sin(x+3)dx). Lower limit: -1, Upper limit: 3 with an error of less than 5 x 10^-4.

    a. Trapezoidal's Rule
    b. Simpsons's Rule
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  2. #2
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    Error for Trap rule is E(t) = k(b-a)^3/(12n^2), in your case E(t) = .0005

    K is the absolute maximum value of the 2nd derivative of the function which you are evaluating the integral of between a and b, which for  \int^1_0 8sin(x+3)\ dx is -8sin(x+3). d^2y/d^2x = -8sin(x+3). Between -1 and 3, this function has a maximum value of y = 8 at x=1.71239. Therefore,  k = 8. Now your error function looks like this:

    E(t) = 8(b-a)^3/(12n^2) ; setting the equation equal to .0005 and solving for n given a = -1 and b=3, you get n = 292.119, and since n is usually a whole number, it is critical to round upwards. Therefore, your minimum n value to achieve an error of <.0005 is n=293. Consequently, if you plug 293 into the original error function you get  E(t) = .000497, so that checks out.

    Error for Simpson's Rule is  E(s) = k(b-a)^5/(180n^4). With simpson's rule, k is calculated by attaining the absolute maximum value of the fourth derivative, being d^4y/d^4x = 8sin(x+3). The absolute maximum value of this function between a = -1 and b = 3 is again y = 8 at x = 1.71239. You may notice the value of the function at that  x = 1.71239 is y = -8, but again we're dealing with absolute values. Your equation is now E(s) = 8(b-a)^5/(180n^4). Setting the equation equal to .0005 given a = -1 and b = 3, you get n = 17.3695, but again rounding up to the next minimum whole number the answer is n = 18. If you plug this back into the error equation for Simpson's rule, your maximum error for n = 18 is .000434, so again that checks out as well.


    Anyways, I hope this helps, and correct me if I am wrong, but these are the numbers I got.
    Last edited by kayman121; February 9th 2010 at 09:35 AM.
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