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Math Help - Riemann Sums

  1. #1
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    Riemann Sums

    hi this is a Riemann sum q:

    Find U, the Riemann Upper Sum for f(x) = x2 on [0,2], using 4 equal sub-intervals


    i know that n=0.5 so i thought the upper limit would be 0.5^2+ 1^2+1.5^2+2^2 which gives 7.5 but the answer is 3.75 so could someone please explain why this is the answer?

    cheers
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  2. #2
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    i realised i forgot to multiply the whole thing by n! so i figured out that question but can someone help me with this one:

    When calculating the Riemann Upper and Lower Sums (U and L) for the function f(x) = x2 on the interval [0,2], what is the smallest number of (equal) sub-intervals needed to make U - L ≤ 0.1 ?
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  3. #3
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    Quote Originally Posted by stobs2000 View Post
    i realised i forgot to multiply the whole thing by n! so i figured out that question but can someone help me with this one:

    When calculating the Riemann Upper and Lower Sums (U and L) for the function f(x) = x2 on the interval [0,2], what is the smallest number of (equal) sub-intervals needed to make U - L ≤ 0.1 ?
    I assume f(x) = x^2
    (fyi, use the caret to signify exponents ... x^2)

    \displaystyle \frac{2}{n}\left[f(x_1) + f(x_2) + ... + f(x_{n})\right] - \frac{2}{n}\left[f(x_0) + f(x_1) + ... + f(x_{n-1})\right] \le 0.1

    finish it.
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    thanks-according to that i get 80 which is the answer but i don't understand where you got 2/n from
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  5. #5
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    Here is another question:

    Given that cos\sqrt{x} decreases on the interval [0,9], estimate the value of \int_{0}^{9}cos\sqrt{x} dx using the Riemann Lower Sum L on this interval with three unequal sub-intervals [0,1], [1,4], [4,9]. Enter your answer correct to two decimal places.


    I know that n for the first 3 terms is 1/3, for the next 3 it is 1 and the next 3 it is 5/3 so to find the lower riemann sum i did:

    (cos\sqrt{0}+cos\sqrt{1/3}+cos\sqrt{2/3})x1/3 +(cos\sqrt{1}+cos\sqrt{2}+cos\sqrt{3})x1 +(cos\sqrt{14/3}+cos\sqrt{19/3}+cos\sqrt{8})x 5/3

    and if i use radians mode on the calculator that gives me -2.49

    ps- sorry about the above- im just learning hoe to use latex editor
    Last edited by stobs2000; September 3rd 2010 at 08:40 PM.
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  6. #6
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    Quote Originally Posted by stobs2000 View Post
    thanks-according to that i get 80 which is the answer but i don't understand where you got 2/n from
    how would you define the width of each subinterval?
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  7. #7
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    Quote Originally Posted by stobs2000 View Post
    Here is another question:

    Given that cos\sqrt{x} decreases on the interval [0,9], estimate the value of \int_{0}^{9}cos\sqrt{x} dx using the Riemann Lower Sum L on this interval with three unequal sub-intervals [0,1], [1,4], [4,9]. Enter your answer correct to two decimal places.
    I have no idea what you are doing in this calculation. There are three subintervals given, so you do not need to calculate n. The first subinterval has a width of 1, the second 3, and the third 5.

    Right Riemann sum using the given subintervals ...

    \cos{{\sqrt{1}} \cdot 1 + \cos{{\sqrt{4}} \cdot 3 + \cos{{\sqrt{9}} \cdot 5 \approx -5.66

    btw ... next time, start a new problem w/ a new thread.
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  8. #8
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    thanks X1000
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