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I am trying to calculate the integral

$$ y(x)=\int^b_a dz\delta(x-w_z) $$

were $\delta$ is the dirac delta function, $a=-3$, $b=3$ and $w$ is a one-dimensional matrix such that for some values of $z$ there is an associated value of $w$.

I think this Integral is related to the Dirac measure.

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    $\begingroup$ Rule of thumb (coming from the underlying mathematics): If different function approximations to the Dirac delta functional give different integral results, then that integral is not well defined. This applies to the 1D case where the singular point of the integrand coincides with an endpoint of integration. $\endgroup$
    – Daniel Lichtblau
    Commented Mar 6, 2021 at 14:26
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    $\begingroup$ @DanielLichtblau The rule comes not only from the underlying mathematics, but from the applications where the abstract mathematical object DiracDelta is used to represent some physical phenomenon. $\endgroup$
    – John Doty
    Commented Mar 6, 2021 at 16:51
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    $\begingroup$ @user64494 That you don't understand it indicates a hole in your mathematics education. We all have those. I suggest you read Bracewell's "The Fourier Transform And Its Applications" if you wish to learn. $\endgroup$
    – John Doty
    Commented Mar 6, 2021 at 19:44
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    $\begingroup$ @user64494 Try approximations that are slightly off-center. Say In[32]:= Integrate[UnitBox[(z - 3 - eps^2/2)/eps]/eps, {z, -3, 3}, Assumptions -> eps > 0 && eps < 1] Out[32]= 3/8 Yet this is a perfectly fine approximation (as eps->0 from above) to a delta function centered at 3. $\endgroup$
    – Daniel Lichtblau
    Commented Mar 6, 2021 at 21:51
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    $\begingroup$ @user64494 Well, if you'd do the experiment with the analog computer, you'd learn how the Dirac delta function works in reality. Remember, Dirac invented it to represent the behavior of physical systems, not as pure mathematics. If your mathematics can't handle it, that means your mathematics struggles with representing reality. Such math can be fascinating, but it's not something I'd use in my work as as scientific instrument designer. On the other hand, DiracDelta is profoundly useful. $\endgroup$
    – John Doty
    Commented Mar 7, 2021 at 14:10

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