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I was wondering if anyone may be able to direct me to the correct materials to understand how the support of a function changes with substitution.

For example, if I have a simple integral with a const a:

$\int_{-\infty}^{a}{(x-a)f(x)dx}$

and I perform a substitution $a-x=y$, then $x=a-y$ and the integral is:

$\int{yf(a-y)dy}$

The resource I have suggests that the new limit of integration is from 0 to infinity, but I cannot figure out why.

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This is just the chain rule at play. If you make the change of variable from $x$ to $y$ (where $y$ is a function of $x$) and the original limits of integration are $a$ and $b$, the new limits of integration are $y(a)$ and $y(b)$, respectively.

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  • $\begingroup$ I'd say it has nothing to do with the chain rule. It's pure algebra; no derivatives come into it. $\endgroup$ – Gerry Myerson Oct 5 '14 at 0:05
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    $\begingroup$ I think of u substitution as chain rule, which is why I mentioned it but perhaps it's not necessary. Feel free to edit it to your liking. $\endgroup$ – Cameron Williams Oct 5 '14 at 0:41

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