Area under curve using Riemann Integrals

Question

Can someone explain why and what this "max" symbol is and why it is necessary, thank you.

I have come across a textbook which states the area under a curve is given by

$$ \lim_{\max\Delta x_k\to 0} \sum_{k=1}^N f(x_k^*)\Delta x_k$$ where $x_{i-1}\le x^*_k\le x_i$ for $\Delta x_k = x_i - x_{i-1}$

I just don't understand the "$\max \Delta x_k \to 0$" part.

Thanks in advance.

Answer 1

$\max\Delta x_k$ really means $\max(\Delta x_1,\ldots,\Delta x_N)$, which is the biggest gap between successive mesh points $x_k$ in the partition of the interval. For convergence, it is not enough that the number of partition points goes to infinity; the mesh points must get close together everywhere, or else you may not be able to resolve some local variation in the integrand.

Answer 2

you have partitioned the domain

$a = x_0<x_1<\cdots<x_{n-1}<x_n=b$

When the partition is sufficiently fine the Riemann sum will converge.

$\max \Delta x \to 0$ is saying your partition is very fine. The largest gap in your partition is arbitrarily small.