Calculus Riemann Sums - why do the partitions have to be of the same size?

Question

To set up Riemann sums for integration, my calculus text will say that the intervals of partition are all of the same size. Isn't it rather the case that they could be any size, as long as they are bounded by a largest partition which goes to zero as we take the limit? Why bother being so explicit about the equalities of each delta x?

Similarly, does it matter that the point within each partition be determined in the same manner each time? As the limit goes to zero, as long as the x-value we choose is somewhere within that segment of the domain...

Answer 1

You are right---they don't! Some texts/examples do so out of convenience, but certainly they should not say that $\Delta x_i$ must be constant across $i$.

Similarly, $x_i$ does not have to be say, a left-endpoint, or right-endpoint, or midpoint of the $i$th subinterval. Any $x_i^*\in [x_i,x_{i+1}]$ will do, and you can vary your method of choosing $x_i^*$ across the various $i$.

As you noted, so long as the norm of the partition tends to $0$, all is well.