Partitioning regularly for the Riemann Sum in Calculus 2 from Openstax

Definition 1 states that every partition of a finite graph is $1$-regular, but Definition 2 has more requirements for the sizes of the partition members. It should be noted that Solution 1 does not utilize regex in its partitioning process and instead looks for an exact match of the given string.


Question:

As I am currently learning mathematics through openstax, I am facing difficulty in comprehending the meaning of the following.

It should be pointed out that we have chosen to use a regular partition in the Riemann sums, although it is not mandatory. Any partition can be utilized to form a Riemann sum. Nonetheless, if we use a nonregular partition to determine the definite integral, simply taking the limit as the number of
subinterval
s approaches infinity is not enough. We need to take the limit as the width of the largest subinterval approaches zero.

My assumption is that the implementation of a standard partition implies that we are dealing with either (1) distinct regions of equal size or (2) a bounded range [a, b] that is the subject of our computation.

I’m not certain of the definition of “partition,” but my guess is that it refers to the range of values between a and b.

I remain perpetually perplexed by the statement, unable to even hazard a guess as to the intended interpretation of the phrase “the largest subinterval goes to zero”. Is there someone who could offer an alternative explanation for this passage?


Solution:

A set of points ${ x_0 = a, x_1, x_2, ldots x_n = b }$ defines a partition of the interval $[a, b]$. Subintervals are defined by each pair of points in the partition. If the partition is regular, then each subinterval has the same width, $Delta x = displaystylefrac{b – a}{n}$. As the number of subintervals approaches infinity, the width of each subinterval approaches zero, and the
sum approaches
approaches the value of the
definite integral
. On the other hand, if the partition is not regular, the width of each subinterval is not necessarily the same. In this case, we need to ensure that the width of the largest subinterval approaches zero to achieve convergence of the Riemann sum to the value of the definite integral it represents.

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