From Partition to Sample Point
A partition divides a closed interval into smaller intervals. To form a rectangle estimate on each of those subintervals, we need a height. The tag supplies the point at which we evaluate the function to choose that height.
The choice is local: each subinterval has its own tag, and that tag must lie in the subinterval it represents. A tag does not have to be the midpoint, and tags chosen for neighboring subintervals need not be different.
The partition points \(x_0,\ldots,x_n\) determine the subintervals and their widths. The tags \(\xi_1,\ldots,\xi_n\) determine where the function is evaluated on those subintervals. These are different roles: a partition point is part of the division, while a tag is selected to represent one particular subinterval.
Because neighboring subintervals share an endpoint, a tag may equal a partition point. For example, the point \(x_i\) can serve as the tag for \([x_{i-1},x_i]\), for \([x_i,x_{i+1}]\), or for both, provided the tag is assigned separately to each subinterval. A tag may also lie strictly inside its subinterval.
How a Tag Selects a Rectangle Height
Suppose \(f\) is defined at every selected tag. On the subinterval \([x_{i-1},x_i]\), the tag \(\xi_i\) selects the function value \(f(\xi_i)\). In a rectangle estimate, this value is used as the constant height across that subinterval. The width is \(x_i-x_{i-1}\), so the signed contribution from that rectangle is the height multiplied by the width.
If \(f(\xi_i)\) is positive, the contribution is positive; if it is negative, the contribution is negative. This is a signed rectangle contribution, not necessarily an ordinary positive geometric area. The tag selects the height, but it does not change the subinterval’s width.
Adding the contributions from all the subintervals gives the finite tagged rectangle sum. For now, the central point is how each term is chosen: use the function value at the tag belonging to that subinterval, then multiply by that subinterval’s width.
The tag choices are made separately. Knowing the tag for one subinterval does not determine the tag for another. In particular, the rule does not require using the same relative position in every subinterval, such as always choosing a left endpoint or always choosing a midpoint.
Worked Examples
Worked Example: Tags Inside Two Subintervals
Let \(f(x)=x^2\) on \([0,2]\). Use the partition \(0<1<2\), with tags \(\xi_1=\frac12\) for \([0,1]\) and \(\xi_2=\frac32\) for \([1,2]\). Find each rectangle contribution and their total.
The first subinterval has width \(1-0=1\). Its tag is \(\frac12\), so its height is
Its contribution is therefore \(\frac14\cdot 1=\frac14\). The second subinterval also has width \(2-1=1\). Its tag is \(\frac32\), so its height is
Its contribution is \(\frac94\cdot 1=\frac94\). Adding the two contributions gives
Check: the selected heights are \(f(\frac12)=\frac14\) and \(f(\frac32)=\frac94\), and both widths are \(1\). Multiplying each height by its width recovers the contributions \(\frac14\) and \(\frac94\), whose sum is \(\frac52\).
Worked Example: A Tag at an Endpoint
Let \(f(x)=3-x\) on \([0,3]\). Take the partition \(0<1<3\), and choose \(\xi_1=0\) for \([0,1]\) and \(\xi_2=3\) for \([1,3]\). Calculate the contributions.
Both tags are valid: \(0\) belongs to \([0,1]\), and \(3\) belongs to \([1,3]\). The first width is \(1-0=1\), and its selected height is
The first contribution is \(3\cdot1=3\). The second width is \(3-1=2\), and its selected height is
The second contribution is \(0\cdot2=0\), so the total is \(3+0=3\). Check: the tag \(3\) selects height zero, so its rectangle contributes zero even though its width is \(2\); the first rectangle contributes \(3\) as calculated.
Worked Example: A Negative Selected Height
Let \(f(x)=x-1\) on \([0,2]\), with partition \(0<1<2\). Choose \(\xi_1=\frac14\) for \([0,1]\) and \(\xi_2=\frac32\) for \([1,2]\). Find the signed contributions and their total.
Each subinterval has width \(1\). For the first subinterval, the selected height is
Thus its contribution is \(-\frac34\cdot1=-\frac34\). For the second subinterval,
so its contribution is \(\frac12\cdot1=\frac12\). The total signed sum is
Check: the negative first height produces a negative contribution, while the positive second height produces a positive one. Their signed total is \(-\frac14\), not the sum of their absolute values.
Why the Product Uses the Tag Value
On a subinterval, a rectangle has one constant height and one width. If the tag \(\xi_i\) is chosen to represent \([x_{i-1},x_i]\), the rule assigns the height \(f(\xi_i)\) to that rectangle. The width is fixed by the partition: it is the distance from the left endpoint to the right endpoint, \(x_i-x_{i-1}\). The rectangle’s signed contribution must therefore be their product.
Use its endpoints \(x_{i-1}\) and \(x_i\); its width is \(x_i-x_{i-1}\).
Evaluate the function at \(\xi_i\), not at a tag belonging to a different subinterval.
The resulting signed contribution is \(f(\xi_i)(x_i-x_{i-1})\).
This reasoning justifies the rectangle-sum rule directly: a rectangle’s area magnitude is its height magnitude times its width, and the sign is determined by the function value. It also explains why tags matter. For a fixed subinterval, changing the tag can change the selected height, and therefore can change that subinterval’s contribution. A tagged sum is determined by both the partition and its tag choices.
This construction is an estimate when a varying function is represented by a constant height on each subinterval. It does not, by itself, claim that a particular tag choice gives an exact accumulated total. The purpose of the tag is to specify precisely which function value is used for the rectangle on that piece.
Common Mistakes and Checks
A frequent error is to use a tag from the wrong subinterval. For example, in the first worked example, the tag \(\frac32\) belongs to \([1,2]\). Using it to calculate the first rectangle would give
but this is not the first subinterval’s contribution. The first subinterval has tag \(\frac12\), so its contribution is instead
The correction is to pair each tag with the subinterval for which it was selected. A reliable check is to write down the endpoints, the tag, the function value, and the width together before multiplying.
Another mistake is to assume that every tag must be a midpoint. A midpoint is a valid choice, but the definition allows any point in the corresponding closed subinterval, including either endpoint. Conversely, a point outside that subinterval is not a valid tag for it, even if it lies somewhere else in the original interval.
Check Your Understanding
For each question, identify the subinterval and its own tag before evaluating the function.
- For the partition \(0<2<5\), which points are allowed as tags for the subinterval \([2,5]\)?
- If \(f(x)=x+2\), the tag is \(\xi=1\), and the corresponding subinterval is \([0,3]\), what height and width determine its rectangle contribution?
- Can the shared partition point \(x_i\) be a tag for both \([x_{i-1},x_i]\) and \([x_i,x_{i+1}]\)? Explain.
- For \(f(x)=-2\) on a subinterval of width \(4\), what is the signed rectangle contribution?
- Why is evaluating \(f\) at a tag from a different subinterval an error?
