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Math Help - Riemann integrability

  1. #1
    Member thaopanda's Avatar
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    Riemann integrability

    Decide which of the functions f_n : [0,1] \rightarrow R with n = 0,1,2, defined by setting:

    f_n(x) :=
    (x^n)sin(\frac{1}{2} if 0 < x \leq 1
    0 if x = 0

    are Riemann integrable on [0,1].

    more help please
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by thaopanda View Post
    Decide which of the functions f_n : [0,1] \rightarrow R with n = 0,1,2, defined by setting:

    f_n(x) :=
    (x^n)sin(\frac{1}{2} if 0 < x \leq 1
    0 if x = 0

    are Riemann integrable on [0,1].

    more help please
    Is that f_n(x)=x^n\sin\left(\frac{1}{2}\right)<br />
?
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  3. #3
    Member thaopanda's Avatar
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    yeah, sorry, I forgot the other parenthesis
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by thaopanda View Post
    yeah, sorry, I forgot the other parenthesis
    Well \sin(1/2) is just a number. Call it k. So you need to know for which n is f_n(x)=kx^n integrable on [0,1].

    For n\geq1, it's continuous and therefore integrable. For n=0, you have one removable discontinuity (at x=0), so it's also integrable.
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  5. #5
    Member thaopanda's Avatar
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    if it was sin(\frac{1}{x}), how would that change the problem?
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  6. #6
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by thaopanda View Post
    if it was sin(\frac{1}{x}), how would that change the problem?
    x^n\sin(1/x) is continuous for all n\geq1. If n=0, the function is still continuous on (0,1] and is therefore integrable.
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