Why Partitions Matter
A rectangular approximation over an interval depends on how that interval is divided into pieces. To describe those pieces precisely, we need a finite list of endpoints in the correct order. Such a list is called a partition of the interval.
Partitions provide the subdivisions on which later integral constructions are based. For now, the main task is to identify the endpoints, the subintervals between consecutive endpoints, and the width of each subinterval. No assumption that the widths are all equal is part of the definition.
Definition of a Partition
Let \(a\) and \(b\) be real numbers with \(a<b\). A partition \(P\) of the closed interval \([a,b]\) is a finite ordered list of real numbers
where \(n\) is a positive integer. The numbers \(x_0,x_1,\ldots,x_n\) are the partition points. They include both endpoints \(a\) and \(b\). The points determine \(n\) subintervals, one between each pair of consecutive partition points.
There are \(n+1\) partition points but only \(n\) subintervals. The first subinterval is \([x_0,x_1]\); the last is \([x_{n-1},x_n]\). The closed subintervals share endpoints where they meet. Their interiors do not overlap, and together they cover the original interval.
The word ordered matters: the list begins at \(a\), ends at \(b\), and moves from left to right without repeating a point. It is not enough to give a collection of numbers that happen to lie in \([a,b]\). Those numbers must be arranged as a strictly increasing list with the required endpoints.
Widths Add to the Length of the Interval
The widths of the subintervals fit together without gaps. Their total must therefore equal the length of the interval. The algebra expresses this by cancellation of consecutive endpoints.
To see why the formula holds, substitute the definition \(\Delta x_i=x_i-x_{i-1}\) into the sum. Each interior endpoint appears once with a positive sign and once with a negative sign, so only the two outer endpoints remain.
The same ordering also explains the coverage property. The first subinterval starts at \(a=x_0\), and each subinterval ends at the point where the next one starts. The last subinterval ends at \(b=x_n\). Consequently there is no gap between consecutive pieces and no part of \([a,b]\) lies outside them.
This is a useful check when constructing a partition: the widths should all be positive, and their sum should match the full interval length. Passing the width-sum check supports a correct setup, though the endpoint order and interval bounds must still be checked as well.
Worked Example: Listing the Pieces of a Partition
Worked Example: Listing the Pieces of a Partition
Consider the partition \(P\) of \([-2,4]\) specified by the ordered endpoints \(-2,-1,1,4\). Identify its subintervals and widths, then verify the total width.
There are four partition points, so there are \(4-1=3\) subintervals. Write the endpoints in order as \(x_0=-2\), \(x_1=-1\), \(x_2=1\), and \(x_3=4\).
The subintervals are formed from consecutive pairs:
Calculate each width by subtracting the left endpoint from the right endpoint:
Each width is positive. Their sum is \(1+2+3=6\). The full interval has length \(4-(-2)=6\), so the widths add to the correct total. The subintervals meet at \(-1\) and \(1\), and the first and last endpoints are \(-2\) and \(4\), as required.
How to Construct a Partition
To construct a partition, decide which breakpoints to use inside the interval, then place them in increasing order between the required endpoints. Every interior breakpoint must be strictly between \(a\) and \(b\), and no two partition points may be equal.
Begin with \([a,b]\), where \(a<b\). The first point must be \(a\), and the last point must be \(b\).
Select any finite number of distinct real numbers strictly between \(a\) and \(b\), if interior points are needed.
List the endpoints and chosen interior points from least to greatest. This gives the partition points \(x_0,\ldots,x_n\).
Pair each point with the next point in the list, then subtract left endpoint from right endpoint to find each width.
Confirm the list begins at \(a\), ends at \(b\), is strictly increasing, and has widths whose sum is \(b-a\).
A partition does not need to use breakpoints that are equally spaced. Its defining requirements are the endpoint conditions and strict ordering. The choice of interior points can be made to suit the subdivision needed for a particular problem.
Worked Example: Constructing a Partition with Interior Points
Worked Example: Constructing a Partition with Interior Points
Construct a partition of \([1,7]\) that includes the interior points \(2.5\) and \(5\). List the subintervals and widths, and check the result.
The required endpoints are \(1\) and \(7\). Both requested interior points lie strictly between them, since \(1<2.5<7\) and \(1<5<7\). Put all four points in increasing order:
There are four partition points and therefore three subintervals:
Calculate their widths one at a time:
The widths are positive, and their total is \(1.5+2.5+2=6\). The interval length is \(7-1=6\). Thus the width sum agrees with the interval length, and the ordered endpoints begin and end at the required values. This verifies the partition.
Partitions on Intervals with Negative Endpoints
Negative endpoints do not change the definition. The endpoints still have to be listed from least to greatest, and widths are still found by subtracting the left endpoint from the right endpoint. Careful use of parentheses is especially helpful when subtracting a negative number.
Worked Example: Partitioning an Interval Across Zero
Construct a partition of \([-5,2]\) using the interior points \(-3\) and \(-1\). Give the subintervals, compute their widths, and verify the total.
First check the order: \(-5<-3<-1<2\). Therefore the partition points are
The consecutive endpoint pairs give the subintervals \([-5,-3]\), \([-3,-1]\), and \([-1,2]\). Their widths are
All three widths are positive, and \(2+2+3=7\). The full interval length is \(2-(-5)=7\). The pieces begin at \(-5\), meet consecutively at \(-3\) and \(-1\), and end at \(2\), so the partition covers the stated interval.
Common Mistakes and How to Correct Them
A common mistake is to subtract endpoints in the wrong order. For example, in the subinterval \([-3,-1]\), an incorrect calculation might be \(-3-(-1)=-2\). That number is negative, but a width cannot be negative. The left endpoint is \(-3\), and the right endpoint is \(-1\), so the correct subtraction is
Another error is to list endpoints without checking their order. For example, \(-5,-1,-3,2\) is not a valid partition list because \(-1\) is greater than \(-3\). Sorting the points gives \(-5,-3,-1,2\), which is strictly increasing and produces positive widths.
Repeated points are not allowed either. A list such as \(0,2,2,5\) contains a zero-width step from \(2\) to \(2\), so it fails the strict-increase condition. A partition must have distinct consecutive endpoints, not merely nondecreasing ones.
Finally, do not confuse the number of points with the number of pieces. A list of \(n+1\) partition points determines \(n\) subintervals: each piece is formed by one adjacent pair. Writing the adjacent pairs explicitly is a reliable way to avoid an indexing error.
Why the Definition Is Useful
The partition definition turns the informal instruction “divide the interval into pieces” into data that can be checked and used precisely. The endpoints say where each piece begins and ends; the widths record the lengths of those pieces. This matters in definite-integral work because approximations over a whole interval are built from contributions associated with its subintervals.
At this stage, the essential skill is not choosing a special kind of partition, but specifying any partition correctly. Once the ordered endpoints are clear, the subintervals and widths follow directly. The endpoint, ordering, positivity, and total-width checks make it possible to catch many setup errors before using a partition in a further calculation.
Check Your Understanding
For each question, keep the endpoints in increasing order and use right endpoint minus left endpoint when calculating a width.
- How many subintervals are determined by the partition points \(0,1,4,6\)?
- Write the subintervals and widths for the partition \([-4,3]\) with partition points \(-4,-2,1,3\).
- Is the list \(2,5,4,8\) a valid partition of \([2,8]\)? Explain why or why not.
- Construct a partition of \([0,10]\) with interior points \(3\) and \(7\), then verify that its widths sum to the interval length.
- Why does a list of \(n+1\) partition points determine \(n\), rather than \(n+1\), subintervals?
