Is this correct notation for the area under a curve using a limit? I've just had a resource tell me the following:
The area under a curve from x=a to x=b is
A=$lim_{\Delta x{\rightarrow}0}\sum_{k=a}^{b}f(x_{k})\Delta x$
I generally see $x_{k}=a+k\Delta x$, but in that case I'm struggling to understand how this is correct if the sum limit is b, shouldn't it instead be the number of subintervals?
ie
$lim_{\Delta x{\rightarrow}0}\sum_{k=1}^{N}f(x_{k})\Delta x$, where $N=\frac{b}{\Delta x}$
either this is wrong or i am misinterpreting what $x_{k}$ is.
Thanks
 A: The first notation is poor and I’d personally says yours is much better. A sum from “$k=a$” to “$k=b$” doesn’t make much sense in the standard interpretation. I can guess what the resource meant however: they intended the summation to be interpreted as a sum not over the indices of the $x_k$ but over the values themselves: start with $x_k=a$, then add the next value of $x_k$, then the next... then add $x_k=b$.
Your notation is much better but because I think it needs explaining to you, I will comment on it. This process is called a Riemann sum. The $x_k$ can be any sequence of partitions (whose differences tend to $0$): the common one to consider is $x_k=a+k\Delta x$ where $\Delta x$ is constant, but it doesn’t need to be that way, and in general you need to understand what the “$x_k$” really are. In that vein, the $\Delta x$ might be different for each $k$: it is more precise to write $(x_{k+1}-x_k)$ (in a “right Riemann sum”) or $(x_k-x_{k-1})$ (in a “left Riemann sum”) instead of $\Delta x$.
