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Math Help - Trapezoid Rule help?

  1. #1
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    Trapezoid Rule help?

    So for this question: http://i57.tinypic.com/15i2m2o.jpg

    I'm having trouble because I get a triangle instead of a trapezoid for the first trapezoid. Because It starts with (0,0). What should I do? What would the area be with the trapezoid rule? And I'm having trouble distinguishing the formulas of the trapezoid rule and midpoint rule. Any help?
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  2. #2
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    Re: Trapezoid Rule help?

    Quote Originally Posted by canyouhelp View Post
    So for this question: http://i57.tinypic.com/15i2m2o.jpg

    I'm having trouble because I get a triangle instead of a trapezoid for the first trapezoid. Because It starts with (0,0). What should I do? What would the area be with the trapezoid rule? And I'm having trouble distinguishing the formulas of the trapezoid rule and midpoint rule. Any help?
    Let's look at the trapezoid rule in its simplest form, where there is only ONE interval:

    $\displaystyle \int_a^bf(x)\ dx \approx (b - a) * \dfrac{f(a) + f(b)}{2}.$

    If f(a) = 0 this reduces to $(b - a) * \dfrac{0 + f(b)}{2} = \dfrac{1}{2} * (b - a) * f(b)$, which is indeed the area of a triangle.

    Now you are using more than one interval for your approximation, but the same logic will apply to subintervals. So for any subinterval where either
    f(a) or f(b) = 0, the associated trapezoid degenerates into a triangle. You did not make a mistake; you discovered something you did not expect. Good job.

    The midpoint formula does have a similarity to the trapezoid rule, but it calls for evaluating the function at only one point in the sub-interval, not two. Again, this is easiest to see if we put each rule into its simplest form, where there is only ONE interval.

    $\displaystyle \int_a^bf(x)\ dx \approx (b - a) * \dfrac{f(a) + f(b)}{2}.$ Trapezoid

    $\displaystyle \int_a^bf(x)\ dx \approx (b - a) * f\left(\dfrac{a + b}{2}\right).$ Midpoint

    In the trapezoid rule we average the values of the function at the endpoints; in the midpoint rule we value the function of the average of the endpoints.

    Does this help?
    Last edited by JeffM; February 24th 2014 at 01:23 PM. Reason: Because I cannot avoid at least one typo
    Thanks from romsek
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  3. #3
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    Re: Trapezoid Rule help?

    A "triangle" can be thought of as a "trapezoid" with one base of length 0.
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