Here's the definition of an integral from Wikipedia:
Given a function $f$ of a real variable $x$ and an interval $[a, b]$ of the real line, the definite integral
$\int_a^b f(x) \, dx$
is defined informally to be the area of the region in the $xy$-plane bounded by the graph of $f$, the $x$-axis, and the vertical lines $x = a$ and $x = b$, such that area above the $x$-axis adds to the total, and that below the $x$-axis subtracts from the total.
Why do they specify a closed interval? Wouldn't using $(a,b)$ make no difference as the contribution to the integral from the endpoints $a$ and $b$ is zero as points have no width?