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Math Help - area using riemann sums

  1. #1
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    area using riemann sums

    I am learning about using riemann sums to calculate area. I came across this example and it doesn't make much sense to me.

    the example is: find the area of the region bounded by y = 16 - x^2 and the x-axis for
    1<x<3

    where I am getting confused is at the end.....

    (30/n)(n)-[8/n^2][1/2(n)(n+1)]-[8/n^3][1/6(n)(n+1)(2n+1)]

    = 70/3 - 8/n -4/3n^2

    I don't understand where the 70/3 comes from. and I predict that -8/n and -4/3n^2 just coming from simplifying the corresponding -[8/n^2][1/2(n)(n+1)] and -[8/n^3][1/6(n)(n+1)(2n+1)]. thank you.
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  2. #2
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    Quote Originally Posted by colorado View Post
    I am learning about using riemann sums to calculate area. I came across this example and it doesn't make much sense to me.

    the example is: find the area of the region bounded by y = 16 - x^2 and the x-axis for
    1<x<3

    where I am getting confused is at the end.....

    (30/n)(n)-[8/n^2][1/2(n)(n+1)]-[8/n^3][1/6(n)(n+1)(2n+1)]

    = 70/3 - 8/n -4/3n^2

    I don't understand where the 70/3 comes from. and I predict that -8/n and -4/3n^2 just coming from simplifying the corresponding -[8/n^2][1/2(n)(n+1)] and -[8/n^3][1/6(n)(n+1)(2n+1)]. thank you.
    To Be honest you notation is hard to read but it looks like you just need to simplify

    If you just extract the constant part from each term you will get

    30-\frac{8}{2}-8\cdot \frac{2}{6}=\frac{70}{3}

    Just carefully simply the whole expression.

    http://www.wolframalpha.com/input/?i=%2830%2Fn%29%28n%29-[8%2Fn^2][%281%2F2%29*%28n%29%28n%2B1%29]-[8%2Fn^3][%281%2F6%29*%28n%29%28n%2B1%29%282n%2B1%29]

    the full result is halfway down the page
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  3. #3
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    the trick is to "cancel the n's" so you have constant terms and things like: (something)/(a power of n) <-- all these types of terms go to 0 as n gets very large, so only the constant terms survive in the limit.

    so (30/n)n = 30, -(8/n^2)(n(n+1)/2) = -(4n^2 + 4n)/n^2 = -4 - 4/n <--- the 4/n term will disappear in the limit.

    and -(8/n^3)(n(n+1)(2n+1)/6) = -(8n^3 + 12n^2 + 4n)/(3n^3) = -8/3 - 4/n - 4/(3n^2) <--- the last two terms will disappear in the limit.

    this leaves us with: 30 - 4 - 8/3 = (90 - 12 - 8)/3 = 70/3.

    (if you carefully add the above, you get 70/3 - 8/n - 4/(3n^2), as your text indicates).
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