Why Equal Subintervals Are Useful
A partition breaks an interval into smaller pieces. When every piece has the same width, the notation for summing contributions becomes especially simple: each sample value is multiplied by one common width. This makes uniform partitions a natural setting for writing and calculating tagged Riemann sums.
The key is to keep three choices distinct. The endpoints \(a\) and \(b\) determine the interval. The positive integer \(n\) determines how many subintervals it is divided into. A tag \(\xi_i\) is then selected inside each subinterval. The common width follows from the first two choices; it does not depend on the tags.
Definition and Common Width
Here \(\Delta x\) is positive because \(b-a>0\) and \(n>0\). There are \(n+1\) partition points but exactly \(n\) subintervals: each consecutive pair \(x_{i-1},x_i\) determines one subinterval. The use of \(i=1,\ldots,n\) for subintervals also explains why the first subinterval has endpoints \(x_0\) and \(x_1\).
The formula for the width follows from the total length. If each of \(n\) subintervals has width \(w\), their combined width is \(nw\). Because they cover \([a,b]\) without gaps or overlaps of positive length, that total is \(b-a\). Therefore \(nw=b-a\), so \(w=(b-a)/n\).
This calculation also verifies that the formula for the partition points really gives equal widths. At the final point, the formula gives \(x_n=a+n\Delta x=a+n(b-a)/n=b\), so the partition reaches the right endpoint exactly.
Tags and the Uniform Tagged Sum
After forming the partition, choose one sample point, or tag, from each subinterval. For the \(i\)th subinterval, the tag must satisfy \(x_{i-1}\leq \xi_i\leq x_i\). A tag may be an endpoint or an interior point. The choice can differ from one subinterval to another.
The indexed sum records one contribution from each subinterval: the function value at its tag, multiplied by that subinterval's width. The tag rule matters because a sample point outside its assigned subinterval would not represent a sample from that piece of the partition.
Common tag choices include left endpoints, right endpoints, and midpoints. With left endpoints, \(\xi_i=x_{i-1}\); with right endpoints, \(\xi_i=x_i\); and with midpoints, \(\xi_i=(x_{i-1}+x_i)/2\). These are particular ways to choose tags, not different formulas for the common width.
Worked Example: Midpoint Tags on an Interval
Worked Example: Midpoint Tags on an Interval
Divide \([1,5]\) into four equal subintervals. Use the midpoint of each subinterval as the tag and calculate the tagged sum for \(f(x)=2x+1\).
First, calculate the common width:
The partition points are \(x_i=1+i\), for \(i=0,1,2,3,4\). Thus \(x_0=1\), \(x_1=2\), \(x_2=3\), \(x_3=4\), and \(x_4=5\). The subintervals are \([1,2]\), \([2,3]\), \([3,4]\), and \([4,5]\).
Their midpoints are \(\xi_1=3/2\), \(\xi_2=5/2\), \(\xi_3=7/2\), and \(\xi_4=9/2\). Evaluate \(f\) at each tag:
Now multiply each value by the common width and add:
Each midpoint lies in its corresponding subinterval, and each subinterval has width \(1\), as required. The computed tagged sum is \(28\).
Worked Example: Right-Endpoint Tags
Worked Example: Right-Endpoint Tags
Divide \([0,3]\) into three equal subintervals. Use right endpoints as tags and calculate the tagged sum for \(f(x)=x^2\).
The common width is
Here \(x_i=0+i\), so the partition points are \(x_0=0\), \(x_1=1\), \(x_2=2\), and \(x_3=3\). The right-endpoint tags are \(\xi_1=1\), \(\xi_2=2\), and \(\xi_3=3\), one from each subinterval. The function values are \(f(1)=1\), \(f(2)=4\), and \(f(3)=9\). Therefore
In indexed form the same calculation is \(\sum_{i=1}^{3}i^2(1)=1^2+2^2+3^2=14\). The tags are exactly the right endpoints, and substituting each into \(f(x)=x^2\) gives the terms used in the sum.
Worked Example: Different Tags in Different Subintervals
Worked Example: Different Tags in Different Subintervals
Divide \([-2,2]\) into four equal subintervals. Choose tags \(\xi_1=-7/4\), \(\xi_2=-1/2\), \(\xi_3=3/4\), and \(\xi_4=7/4\), and calculate the tagged sum for \(f(x)=x^2+1\).
The common width is
The partition points are \(x_i=-2+i\), giving \(-2,-1,0,1,2\). The subintervals are \([-2,-1]\), \([-1,0]\), \([0,1]\), and \([1,2]\). Each chosen tag belongs to its assigned subinterval.
Evaluate the function at the tags:
Express \(5/4\) with denominator \(16\), then add the four contributions and multiply by the common width:
The result is \(175/16\), or \(10.9375\). The calculation illustrates that the tags need not all follow one rule: what matters for this sum is that each \(\xi_i\) lies in its assigned subinterval.
A Common Width Does Not Mean a Common Tag
Uniformity describes the lengths of the subintervals, not the selection of sample points. Every subinterval has width \(\Delta x\), while its tag can be chosen separately. Consequently, the width can be factored out of the sum, but the function values \(f(\xi_i)\) generally cannot be treated as equal.
A Frequent Indexing Mistake
A common error is to divide the interval length by \(n-1\) when there are \(n\) subintervals. That denominator counts the spaces between a list of \(n\) points, but a partition into \(n\) subintervals has \(n+1\) partition points. For example, dividing \([1,5]\) into four subintervals gives width \((5-1)/4=1\), not \((5-1)/(4-1)=4/3\).
The incorrect width \(4/3\) would make four subintervals have total width \(4(4/3)=16/3\), which is not the interval length \(5-1=4\). The correct width passes the check \(4(1)=4\). As a reliable check, multiply the proposed common width by the number of subintervals; the result must be \(b-a\).
Why the Notation Matters
The uniform-partition notation keeps the geometry and the summation aligned. The index \(i\) identifies a subinterval; \(x_{i-1}\) and \(x_i\) are its endpoints; \(\xi_i\) is its tag; and \(\Delta x\) is its width. Once these pieces are consistent, a tagged sum can be written without listing every rectangle separately.
This consistency is particularly useful when the number of subintervals changes. For each chosen positive integer \(n\), the width is determined by the fixed interval, and the endpoint formula generates all partition points. The same indexed form then records all contributions using one expression.
Check Your Understanding
Answer each question using the definitions and notation from this tutorial.
- What is the common width when \([2,8]\) is divided into six equal subintervals?
- Write the partition points \(x_i\) for a uniform partition of \([0,4]\) into \(n\) subintervals.
- For a partition into five subintervals, how many partition points are there, and what are their indices?
- State the condition that a tag \(\xi_i\) must satisfy for the \(i\)th subinterval.
- If the common width is \(3/2\), the tags are \(\xi_1,\ldots,\xi_n\), and the function is \(f\), write the tagged sum in sigma notation.
