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Math Help - Riemann Sums

  1. #1
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    Last edited by ski4life912; April 10th 2008 at 10:15 PM.
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  2. #2
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    Come to think of it maybe some kind of parabola or something where it changes direction and areas cancel out?
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  3. #3
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    Did you type the question correctly or am I missing something ? There are several functions which are continuous over [-1,2] with partitions of that same interval used to calculate the areas under them.
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    Well right, the general question was do they exist? if so give an example if not why not, at first I didn't think they existed but now im just thinking of examples that fit that.
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    Any linear equation would work really. For example, consider y = -x + 3 over the interval [-1, 2].

    Length of each subinterval:
    \Delta x = \frac{2 - (-1)}{n} = \frac{3}{n}

    Consider your partition with left sample points:
    x_{0} = - 1 \:\: < \:\: x_{1} = -1 + \frac{3}{n} \:\: < \:\: x_{2} = -1 + 2\cdot \frac{3}{n} \:\: < \:\: ... \:\: <  {\color{blue}x_{i} = -1 + \frac{3i}{n}} \:\: < \:\: ... \:\: < \:\: x_{n} = -1 + \frac{3n}{n} = 2

    So your Riemann sum:
    \lim_{n \to \infty}\sum_{i = 1}^{n} f\left(x_{i}^{*}\right) \Delta x

    x_{i}^{*} = x_{i}

    etc. etc.
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