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Math Help - integral inner product

  1. #1
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    integral inner product



    Here's the solution provided:


    I don't understand how they got that answer! This is my own attempt:

    Since f is a piece-wise function I divided the interval to two and calculated the inner product like this:

    \left\langle f, sin(x) \right\rangle = \int^{0}_{-\pi} -sin(x)+\int^{\pi}_{0} sin(x)= cos(0)-cos(\pi)-(-cos(\pi)-cos(0))=-2

    So, why is my method and my answer wrong? And could anyone please explain the method used in the model answers?
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  2. #2
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    -\int_{-\pi}^{0}sin(x)dx+\int_{0}^{\pi}sin(x)dx=cos(x)]_{-\pi}^{0}-cos(x)]_{0}^{\pi}
    =cos(0)-cos(-\pi)-cos(\pi)+cos(0)=1-(-1)-(-1)+1=1+1+1+1=4
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  3. #3
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    The method used in the mode proof is this: If a \geq 0 and g : [-a,a] \to \mathbb{R} is integrable then:
    \int_{-a}^a g(x) \mathrm{d} \! x = \begin{cases} 2 \int_0^a g(x) \mathrm{d} \! x & \text{if } g \text{ is even.} \\ 0 & \text{if } g \text{ is odd.} \end{cases}
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  4. #4
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    Quote Originally Posted by Giraffro View Post
    The method used in the mode proof is this: If a \geq 0 and g : [-a,a] \to \mathbb{R} is integrable then:
    \int_{-a}^a g(x) \mathrm{d} \! x = \begin{cases} 2 \int_0^a g(x) \mathrm{d} \! x & \text{if } g \text{ is even.} \\ 0 & \text{if } g \text{ is odd.} \end{cases}
    Thanks. But how did they figure out that the given function f(x) is odd? Could you please explain that?
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  5. #5
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    Quote Originally Posted by demode View Post
    Thanks. But how did they figure out that the given function f(x) is odd? Could you please explain that?
    Ah, I seem to have glanced over something. Well \forall x \in (0,\pi], f(x) = -f(-x) and integrating over a point doesn't contribute to any integrals, so you could swap f for:
    f(x) := \begin{cases} -1 & \text{if } -\pi \leq x < 0 \\ 0 & \text{if } x = 0 \\ 1 & \text{if } 0 < x \leq \pi \end{cases}

    Then f would be odd and so f(x) \sin(x) is odd, since \sin(x) is even and the product of 2 odd functions is even. But explicit calculation is just as fine.
    Last edited by Giraffro; April 17th 2010 at 12:22 AM. Reason: Mistake
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