Integrating on open vs. closed intervals What is one difference in the values of
$$\int\limits_{\left[0,1\right]}y\, dx$$$$\int\limits_{\left(0,1\right)}y\, dx$$
and how would you calculate the values? For the sake of simplicity, let $y=x$. Conceptualizing integration as the area bounded by the function, the $x$-axis and the limits of integration, the latter should be smaller.

This is not a definitive answer, considering an answer have already accepted almost 2 years ago, but I believe it should be noted that in the case where there exists a Dirac delta function or any similarly defined special functions inside the integrand, the integral over some 'points' become non-zero by definition.
Using this fact, we can prove that the claim that these two integrals are equal leads to a contradiction if one or both of the limits of the interval are included a specially defined function. In this case, inclusion of a particular point will change the answer of the integration by a finite amount. Choosing this point from the limits of the interval, integration over an open interval will yield a different result than integration over an open interval by that finite amount.
So, integration on open or closed intervals can yield different results if there is a special function in the integrand such as Dirac delta.

Disclaimer: Notion of integrating δ(x) over an interval [0,a] might be undefined on its own. If anybody have any information in this regard, please edit the answer accordingly.

See also: Dirac delta integral with $\delta(\infty) \cdot e^{\infty}$
