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Math Help - Upper and Lower Riemann Sums

  1. #1
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    Upper and Lower Riemann Sums

    Compute the upper and lower Riemann sums and the integral from zero to 1 of f(x)dx, where the partition is

    P= {0, 2/5, 1/2, 3/5, 1}

    Thanks.
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  2. #2
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    Quote Originally Posted by taypez View Post
    Compute the upper and lower Riemann sums and the integral from zero to 1 of f(x)dx, where the partition is

    P= {0, 2/5, 1/2, 3/5, 1}

    Thanks.
    What is f(x)?
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    f(x) = 1-x^2
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    Quote Originally Posted by taypez View Post
    Compute the upper and lower Riemann sums and the integral from zero to 1 of f(x)dx, where the partition is

    P= {0, 2/5, 1/2, 3/5, 1}

    Thanks.
    f(x) = 1 - x^2

    f(x) is decreasing from 0 to 1. hence, for the given partition:

    U(P,f) = 1\cdot \frac{2}{5} + \frac{21}{25} \cdot \frac{1}{10} + \frac{3}{4} \cdot \frac{1}{10} + \frac{16}{25} \cdot \frac{2}{5} = \frac{163}{200}

    L(P,f) = \frac{21}{25} \cdot \frac{2}{5} + \frac{3}{4} \cdot \frac{1}{10} + \frac{16}{25} \cdot \frac{1}{10} + 0 \cdot \frac{2}{5} = \frac{19}{40}
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    Can you give me an example of a couple calculations on how you arrived at these numbers for U(f,P)? I came up with some different ones and I can't find my mistake.

    Thanks.
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    Quote Originally Posted by taypez View Post
    Can you give me an example of a couple calculations on how you arrived at these numbers for U(f,P)? I came up with some different ones and I can't find my mistake.

    Thanks.
    Because U(f,P) is calculated by taking the supremum (largest value) on each interval. So since the function was decreasing the largest value is the left-endpoint.
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