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Math Help - Reimann integral proof by induction

  1. #1
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    Lightbulb Reimann integral proof by induction

    Question:

    Suppose that f(x)=x for all x∈[0,b]. Show that f is integrable and that ∫₀^{b}f(x)dx=((bē)/2)


    Hint for the proof:
    For each n∈ℕ consider the partition P_{n}={0,(b/n),((2b)/n),...,(((n-1)b)/n),b}.

    Induction after this but it can't be at n=0 because we end up dividing by 0?
    Need some help on this one. Using Reimann.

    Thanks.
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  2. #2
    Super Member girdav's Avatar
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    Re: Reimann integral proof by induction

    It should be \mathbb{N}^*, and I think it's only a typo. We can compute the integral thanks to this partition, but before that we have to show that the function is Riemann-integrable.
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  3. #3
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    Re: Reimann integral proof by induction

    Quote Originally Posted by CountingPenguins View Post
    Question:

    Suppose that f(x)=x for all x∈[0,b]. Show that f is integrable and that ∫₀^{b}f(x)dx=((bē)/2)


    Hint for the proof:
    For each n∈ℕ consider the partition P_{n}={0,(b/n),((2b)/n),...,(((n-1)b)/n),b}.

    Induction after this but it can't be at n=0 because we end up dividing by 0?
    Need some help on this one. Using Reimann.

    Thanks.
    Hi CountingPenguins,

    There is no strict rule as to include 0 in the natural number set or not (Refer: Natural number - Wikipedia, the free encyclopedia). In this context they must have meant the natural numbers as the positive integers, because including 0 would be meaningless.
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