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Math Help - Shifted fcn...

  1. #1
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    Shifted fcn...

    Given:

    x(t) = u(t) - u(t-10)

    calculate:

    Int(x^2)tdt


    Am I thinking about this correctly... u(t) is 0 from neg. infinity to 0, and 1 after. u(t-10) is the same thing just shifted to the right to "turn on" at 10. So in effect we have a rectangle of amplitude 1 with bounds from 0 to 10. So our integral becomes:

    int (1)^2 dt (with bounds of 0 and 10), which = 10?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    your thinking won't be true in general (and we don't even have a definite integral here). Take u(t) = \sin t, for instance. I think we need more info here, if you have some definite answer to get to, as opposed to an expression.
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  3. #3
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    I'm assuming that u(t) is intended to notate the Heaviside step function, not an arbitrary function.

    In any case, could you clarify whether this is a definite or indefinite integral? If we take it as written, it would be an indefinite integral, i.e.,

    \int x^2(t) \, dt

    As you observed, you could define x^2(t) as a piecewise function like so:

    x^2(t) = \begin{cases} 0, & \mbox{if } t<0 \\ 1, & \mbox{if } 0 \le t < 10 \\ 0, & \mbox{if } t \ge 10 \end{cases}

    Then we could find the anti-derivative using piecewise techniques.

    If instead you are calculating a definite integral whose bounds cover the 0 to 10 range, then your work is correct.
    Last edited by drumist; July 1st 2010 at 03:34 AM.
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