What is the name of the vertical bar in $(x^2+1)\vert_{x = 4}$ or $\left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$? I've always wanted to know what the name of the vertical bar in these examples was:

$f(x)=(x^2+1)\vert_{x = 4}$ (I know this means evaluate $x$ at $4$)
$\int_0^4 (x^2+1) \,dx = \left.\left(\frac{x^3}{3}+x+c\right) \right\vert_0^4$ (and I know this means that you would then evaluate at $x=0$ and $x=4$, then subtract $F(4)-F(0)$ if finding the net signed area)

I know it seems trivial, but it's something I can't really seem to find when I go googling and the question came up in my calc class last night and no one seemed to know.
Also, for bonus internets; What is the name of the horizontal bar in $\frac{x^3}{3}$? Is that called an obelus?

Jeff Miller calls it "bar notation" in his Earliest Uses of Symbols of Calculus (see below). The bar denotes an evaluation functional, a concept whose importance comes to the fore when one studies duality of vector spaces (e.g. such duality plays a key role in the Umbral Calculus).

The bar notation to indicate evaluation of an antiderivative at the two limits of integration was first used by Pierre Frederic Sarrus (1798-1861) in 1823 in Gergonne’s Annales, Vol. XIV. The notation was used later by Moigno and Cauchy (Cajori vol. 2, page 250). 

Below is the cited passage from Cajori


