Why Partitions Matter
A definite integral describes accumulated change across an interval. To organize that interval into manageable pieces, we divide it at selected points. The resulting pieces are called subintervals, and each has a width: the distance from its left endpoint to its right endpoint.
That width is a basic part of describing how an interval has been divided. A partition might split the interval into equal-width pieces, or it might use pieces of different widths. Recognizing which situation applies—and calculating the widths correctly—sets up later work with integral approximations.
What Is a Partition?
Let \(a<b\). A partition of the interval \([a,b]\) is a finite, ordered list of points that begins at \(a\), ends at \(b\), and increases strictly from one point to the next. Write the points as \(x_0,x_1,\ldots,x_n\), where \(n\) is the number of subintervals.
Each consecutive pair of partition points determines one subinterval: \([x_{i-1},x_i]\), for \(i=1,\ldots,n\). Its width is the difference between its right endpoint and its left endpoint.
The notation \(\Delta x_i\) means the width of a particular subinterval; it does not mean that every subinterval has the same width. The subscript matters: \(\Delta x_1,\Delta x_2,\ldots,\Delta x_n\) may be different values. For an equal-width partition, all those values happen to be the same.
The partition points must be listed in increasing order. This ensures that the subintervals proceed from left to right, cover the full interval, and meet at their shared endpoints. They do not leave gaps or overlap in their interiors.
Calculating Subinterval Widths
For each subinterval, subtract its left endpoint from its right endpoint. Keep the endpoints in the order specified by the partition. For example, the width of \([2,5]\) is \(5-2=3\), not \(2-5=-3\). A width represents a positive length.
For a partition with several subintervals, calculate each width separately. Do not assume the widths match simply because the partition contains a fixed number of pieces. The locations of the interior partition points determine the individual widths.
Worked Example: Finding Widths in a Partition
Consider the partition of \([-2,4]\) with points \(x_0=-2\), \(x_1=-\frac12\), \(x_2=1\), and \(x_3=4\). There are three subintervals. Find their widths and check that they cover the interval’s total length.
For the first subinterval, the endpoints are \(-2\) and \(-\frac12\):
For the second subinterval:
For the third subinterval:
The widths are \(\frac32,\frac32,3\). Their sum is \(\frac32+\frac32+3=6\). The full interval has length \(4-(-2)=6\), so the width check agrees. The first two widths match, but the third does not; therefore this is not an equal-width partition.
Equal-Width Partitions
A partition is an equal-width partition when every subinterval has the same width. If \([a,b]\) is divided into \(n\) equal subintervals, their common width is the full interval length divided by \(n\).
The partition points can be generated by starting at \(a\) and adding this width repeatedly. The formula for the \(i\)th point, with \(i=0,1,\ldots,n\), is \(x_i=a+i\Delta x\). At \(i=0\) this gives \(x_0=a\); at \(i=n\), it gives \(x_n=a+n\frac{b-a}{n}=b\). Each step between consecutive points has the same length.
Worked Example: Dividing an Interval into Equal Widths
Divide \([1,7]\) into four equal subintervals. Find the common width and list every partition point.
The interval length is \(7-1=6\). With \(n=4\), the common width is
Start at \(x_0=1\) and add \(\frac32\) each time:
The partition points are \(1,\frac52,4,\frac{11}{2},7\). Check the widths by subtracting consecutive points:
All four widths equal \(\frac32\), and their sum is \(4\cdot\frac32=6\), the interval length. Both checks confirm the partition.
Unequal-Width Partitions
A partition has unequal widths if its subintervals are not all the same width. It is enough for two widths to differ; the remaining widths can match one another or differ as well. “Unequal” does not mean that every width must be distinct.
For example, widths \(1,2,2\) are unequal as a collection because not all three values match. Widths \(1,2,3\) are also unequal. In both cases, the widths must be positive, and their sum must equal the length of the full interval.
Worked Example: Checking an Unequal-Width Partition
Consider \([0,10]\) with partition points \(0,1,4,\frac{13}{2},10\). Find each width, determine whether the widths are equal, and verify their total.
There are four subintervals. Subtract each left endpoint from the next point:
The widths are \(1,3,\frac52,\frac72\). They are positive and not all equal, so this is an unequal-width partition. Their sum is
The full interval length is \(10-0=10\), confirming that the widths add up correctly.
Why the Widths Add to the Full Length
For any partition, the subintervals fit together consecutively. The width of each piece is a difference of neighboring partition points. When the differences are added, the interior points cancel, leaving only the last endpoint minus the first. This is why the widths must sum to the full interval length, whether they are equal or unequal.
Here is the complete algebraic justification. Substitute \(\Delta x_i=x_i-x_{i-1}\) into the sum:
Every interior partition point appears once with a positive sign and once with a negative sign, so those terms cancel. The first point is \(x_0=a\), and the last is \(x_n=b\), which proves the result. The identity is useful as a check, but it does not by itself show whether a partition is equal-width: different lists of positive widths can have the same total.
A Common Width Mistake
A frequent error is to subtract endpoints in the wrong order, producing a negative number and treating it as a width. Suppose a subinterval runs from \(\frac12\) to \(3\). The incorrect calculation reverses the endpoints:
This is not the subinterval’s width. The right endpoint is \(3\), so subtract the left endpoint \(\frac12\) from it:
The width is \(\frac52\), a positive length. In general, follow the ordered partition and calculate \(x_i-x_{i-1}\). Do not take an absolute value to conceal reversed partition points: a valid partition already lists its points in increasing order.
Why the Distinction Matters
Equal and unequal partitions describe different ways of dividing the same interval. Equal widths can make repeated calculations more uniform. Unequal widths allow a partition to use smaller pieces in some regions and larger pieces in others. Regardless of the choice, recording every width correctly is essential when the subintervals are used to describe accumulation.
Before using a partition, check three things:
- The first point is the interval’s left endpoint, and the last point is its right endpoint.
- The points are listed in strictly increasing order, so every width is positive.
- The widths sum to \(b-a\); equal-width status is determined by comparing the individual widths, not just their total.
Check Your Understanding
Answer each question using the definitions and checks from this tutorial.
- For the partition \(-1,2,5,8\) of \([-1,8]\), find every width and determine whether the partition has equal widths.
- Divide \([2,14]\) into six equal subintervals. What is the common width?
- A partition of \([0,9]\) has widths \(2,2,5\). Is it equal-width? Does its total width match the interval length?
- Explain why the widths of a partition always add to \(b-a\).
- What is the width of the subinterval \([\frac34,\frac94]\), and which endpoint should be subtracted from which?
