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Step Function

  1. Aug 4, 2006 #1
    Suppose [tex] U(x) [/tex] is the step function and [tex] \delta(x) [/tex] is its derivative. Find [tex] \int^{6}_{-2}(x^{2}-8)\delta(x)\dx [/tex]. I know [tex] \delta(x) = 0 [/tex] for all [tex] x [/tex] except [tex] x = 0 [/tex]. So at [tex] x = 0 [/tex] [tex] v(x) = - 8 [/tex]. After this step, how do we get [tex] \int_{-2}^{2}(x^{2}-8)\delta(x)\dx +\int^{6}_{2}(x^{2}-8)\delta(x)\dx = -8 + 0 [/tex]. How do you get the limits of integration?

  2. jcsd
  3. Aug 5, 2006 #2


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    If by step function you mean

    [tex]U(x)=\left\{\begin{array}{cc}0,&\mbox{ if }x\leq 0\\1, & \mbox{ if } x\geq 0\end{array}\right.[/tex]​

    (a.k.a. the Heaviside step function) then [tex]\frac{d}{dx}U(x) = \delta (x)[/tex] is a Dirac delta function which has the so-called snifting property, that is

    [tex]\int_{-\infty}^{\infty}f(x)\delta (x) \, dx = f(0)[/tex]​

    so the limits of integration don't matter (so long as they contain zero) and the value of the integral is -8.
  4. Aug 5, 2006 #3
    Remember that you can split up the integral. So,

    [tex] \int^{6}_{-2}(x^{2}-8)\delta(x)\dx = \int_{-2}^{0}(x^{2}-8)\delta(x)\dx = \int_{0}^{6}(x^{2}-8)\delta(x)\dx [/tex]

    [tex] \int^{6}_{-2}(x^{2}-8)\delta(x)\dx = \int_{-2}^{0}(x^{2}-8)\delta(x)\dx = \int_{0}^{1}(x^{2}-8)\delta(x)\dx =\int_{1}^{6}(x^{2}-8)\delta(x)\dx[/tex]

    or however you want (within the bounds of the original limits, etc...).

    So the integral equals 0 for some limits of integration, it will hit 0 and the sifting property kicks in, and then the integral equals 0 again.
    Last edited: Aug 5, 2006
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