Is this Integration From First Principles notation standard outside of A Level Maths? I teach A Level Maths in the UK. We are required to do some 'introduction' to integral from first principles as part of the specification (link, page 25 is the interesting part).
In a previous exam question (Paper 2, June 2018), this was essentially the question:

Suppose we have the curve $y= \sqrt{x}$


The point $P(x,y)$ lies on the curve. Consider a rectangle with height $y$ and width $\delta x$.


Calculate $\displaystyle \lim_{\delta x \rightarrow 0}\sum_{x = 4}^9 \sqrt{x} \,\delta x$

The answer involves us recognising $$ \lim_{\delta x \rightarrow 0}\sum_{x = 4}^9 \sqrt{x} \, \delta x = \int_4^9 \sqrt{x} \, dx$$ and evaluating the integral.
Is this notation standard?
To me, it doesn't make sense. How can you have $x=4$ to $9$ as the limits on the sum, for example? A sum only works over integral values.
In addition, one could easily give meaning to the limit as $\displaystyle \lim_{\delta x \rightarrow 0}\sum_{x = 4}^9 \sqrt{x} \, \delta x = \lim_{\delta x \rightarrow 0}( \sqrt{4}(\delta x) + \sqrt{5} (\delta x) + \cdots + \sqrt{9} (\delta x))$ which should be $0$ as $\delta x \rightarrow 0$.
The reason I am so confused is that this question and notation appears on a Pearson A Level Maths Exam paper. It is a regulated qualification. There are very qualified people out there that have deemed this to make sense and be used to assess the understanding of thousands of A Level students in the UK.
In my experience as a UK mathematician, this isn't standard notation. The aim is to convey $x$'s step size is $\delta x$ rather than $1$. I'd recommend writing the limit as $\lim_{\delta x\to0}\sum_{k=0}^{5/\delta x}\sqrt{4+k\delta x}\delta x$ - no, wait, make it $\lim_{N\to\infty}\sum_{k=0}^N\sqrt{4+5k/N}(5/N)$. This limit is the stated integral; in fact, if we set the maximum $k$ to $N-1$ instead (which won't change the limit), it's a Riemann sum.
Just change the equals sign to an arrow. Altogether, it should sufficiently demonstrate the intention to change a discrete sum to a more accurate continuous sum. The equals sign is indeed improper, so a simple arrow addresses this without exceeding A-Level needs. It is also reasonable ink usage, for a generally trivial incongruity.