I had some teachers in high school which uses an expression composed of a function $f(x)$ followed by a vertical bar followed by an lower value $a$ as subscript and an upper value $b$ as superscript. This expression than expands to $f(b)-f(a)$. That is, this notation is defined as:
$$f(x) \Bigr|_{a}^{b} = f(b)-f(a).$$
They use it mainly for definite integral: $$F(x) \Bigr|^{b}_{a} = F(b)-F(a)= \int_a^b f(x)\,\mathrm{d}x,$$ but also for other things:
- $n^x \Bigr|_{a}^b = \dfrac{n^b}{n^a}$.
- $\lim(x) \Bigr|_{a_n}^{b_n} = \lim b_n - \lim a_n = \lim\limits_{n\to\infty} (b_n-a_n)$.
- $\displaystyle \int_0^1 x\,\mathrm{d}x \Bigr|_a^b = \int_0^1 b\,\mathrm{d}x - \int_0^1 a\,\mathrm{d}x = \int_0^1 (a-b)\,\mathrm{d}x$.
What is the name of this notation? Is it in worldwide common usage or is just a shorthand they use?
