From a Tagged Partition to a Sum
A tagged Riemann sum turns a partition of an interval into a numerical estimate of accumulated signed area. Its construction has a simple rule: for each subinterval, evaluate the function at that subinterval’s chosen tag, multiply that value by the subinterval’s width, and then add the resulting products.
The pairing matters. A function value from one subinterval must be multiplied by the width of that same subinterval. The partition may have equal or unequal widths, and a tag may be any point in its own subinterval, including an endpoint.
Each term has two components: \(f(t_i)\), the function value at the tag, and \(\Delta x_i\), the width of the corresponding subinterval. The full sum has one term for every subinterval. The symbol \(t_i\) identifies the tag on the \(i\)th subinterval; it does not have to be its midpoint.
This is a finite sum formed from a particular partition and particular tags. It is not, by itself, a definite integral. Different choices of partition or tags can give different sums. The construction requires only that each tag belongs to its corresponding subinterval and that the function value at each tag is defined.
Why the Terms Represent Rectangles
On the \(i\)th subinterval, imagine a rectangle whose horizontal base runs from \(x_{i-1}\) to \(x_i\). Its base length is \(\Delta x_i\), and its signed height is \(f(t_i)\). The signed area of this rectangle is its height multiplied by its base length, so it is exactly \(f(t_i)\Delta x_i\).
Adding the signed areas of all these rectangles gives the tagged Riemann sum. This explains why each value must be paired with its own width: each product describes one specific rectangle. If the function value is negative, the rectangle lies below the horizontal axis and its signed area is negative. A sum of such terms accounts for cancellation between rectangles above and below the axis.
Worked Example: A Partition with Unequal Widths
Worked Example: Pair Each Function Value with Its Width
Let \(f(x)=2x+1\) on \([0,3]\). Use the partition points \(x_0=0\), \(x_1=1\), and \(x_2=3\), with tags \(t_1=0.5\) and \(t_2=2\). Write and calculate the tagged Riemann sum.
There are two subintervals: \([0,1]\) and \([1,3]\). Their widths are $$ \Delta x_1=x_1-x_0=1-0=1, \qquad \Delta x_2=x_2-x_1=3-1=2. $$
Each tag belongs to its assigned subinterval: \(0.5\in[0,1]\) and \(2\in[1,3]\). Evaluate the function at both tags: $$ f(t_1)=f(0.5)=2(0.5)+1=2, \qquad f(t_2)=f(2)=2(2)+1=5. $$
Pair the first value with the first width and the second value with the second width: $$ f(t_1)\Delta x_1+f(t_2)\Delta x_2 =2(1)+5(2) =2+10 =12. $$
The sum is \(12\). As a check, the two rectangle contributions are \(2\) and \(10\), and \(2+10=12\). The different widths are reflected in the different multipliers; using a common width here would not represent this partition.
Working with Numerical Data
A formula for \(f\) is not necessary if the function values at the chosen tags are supplied. The same construction applies to a table: find each subinterval’s width, read the value corresponding to its tag, multiply the matched pair, and add the products.
In applications, the function value may carry units. For example, if \(f\) is a rate measured in litres per minute and the horizontal variable is time in minutes, then \(f(t_i)\Delta x_i\) is measured in litres. Keeping the width attached to the right rate both preserves the calculation and gives the term the correct units.
Worked Example: Rates Given at Tags
A flow rate \(r(t)\), measured in litres per minute, is recorded at three tags. The partition points are \(0,2,5,6\) minutes, the tags are \(1,3,5.5\) minutes, and the recorded rates are \(r(1)=4\), \(r(3)=-1\), and \(r(5.5)=3\) litres per minute. Construct and calculate the tagged sum.
The subintervals are \([0,2]\), \([2,5]\), and \([5,6]\). Their widths are $$ \Delta t_1=2-0=2,\qquad \Delta t_2=5-2=3,\qquad \Delta t_3=6-5=1 $$ minutes. Each listed tag lies in its corresponding subinterval.
Multiply each recorded rate by its own time width: $$ r(1)(2)+r(3)(3)+r(5.5)(1) =4(2)+(-1)(3)+3(1) =8-3+3 =8. $$
The tagged sum is \(8\) litres. The units check: each term is \((\text{litres per minute})(\text{minutes})=\text{litres}\). The negative rate contributes \(-3\) litres, so the signed contributions add to \(8\) litres. Checking the arithmetic, \(8-3=5\) and \(5+3=8\).
Reading the Sum from a Graph
On a graph, each subinterval determines a rectangle’s base, and the function value at its tag determines the rectangle’s height. To read or construct the sum graphically, locate the tag on the horizontal axis, read the function’s height there, and use that height across the tag’s subinterval. The tag determines the rectangle’s height even when the function itself changes height across the subinterval.
If the tag’s function value is positive, that rectangle contributes positively. If it is negative, it contributes negatively. A tag where the function is zero gives a zero contribution, regardless of the width of the subinterval. The rectangles need not follow the curve between tags: the sum uses one chosen height on each subinterval.
Worked Example: Signed Rectangles on a Graph
Consider the line \(f(x)=x-2\) on \([0,4]\), partitioned at \(0,1,3,4\). Choose tags \(t_1=0.5\), \(t_2=2.5\), and \(t_3=3.5\). Construct the sum and interpret the signs of its terms.
The three subintervals are \([0,1]\), \([1,3]\), and \([3,4]\). Their widths are $$ \Delta x_1=1-0=1,\qquad \Delta x_2=3-1=2,\qquad \Delta x_3=4-3=1. $$
The tags belong to their respective subintervals. The function values are $$ f(0.5)=0.5-2=-1.5,\qquad f(2.5)=2.5-2=0.5,\qquad f(3.5)=3.5-2=1.5. $$
Now match each height with its own width: $$ (-1.5)(1)+(0.5)(2)+(1.5)(1) =-1.5+1+1.5 =1. $$
The first rectangle is below the horizontal axis and contributes \(-1.5\). The other two are above it and contribute \(1\) and \(1.5\). Their signed contributions total \(1\), since \(-1.5+1+1.5=1\). In particular, the middle rectangle has height \(f(2.5)=0.5\) across the whole interval \([1,3]\); the sum does not use the line’s varying height at every point of that interval.
A Construction Procedure
For a new problem, it helps to keep the partition, tags, values, and widths distinct until the final addition. This prevents an arithmetic error from hiding a setup error.
Read consecutive partition points to identify each interval \([x_{i-1},x_i]\).
Confirm that each \(t_i\) belongs to its assigned interval.
Calculate \(\Delta x_i=x_i-x_{i-1}\), and evaluate \(f(t_i)\) or read the supplied value.
Form one product \(f(t_i)\Delta x_i\) per interval, then add them with their signs.
Common Mistake: Mismatching a Tag and a Width
A frequent error is to use the correct function values and widths but pair them in the wrong order. For example, in the first worked example, the values are \(2\) and \(5\), and the widths are \(1\) and \(2\). A wrong calculation might pair \(2\) with width \(2\) and \(5\) with width \(1\): $$ 2(2)+5(1)=4+5=9. $$
This calculation is not the tagged sum for the stated partition and tags. The value \(2\) comes from the tag in \([0,1]\), whose width is \(1\); the value \(5\) comes from the tag in \([1,3]\), whose width is \(2\). The correction is $$ 2(1)+5(2)=2+10=12. $$
A reliable check is to write each interval, tag, value, and width together before multiplying. Also check the signs and units of the terms. These checks can catch errors even when the final addition is correct.
What the Sum Does—and Does Not—Say
A tagged Riemann sum gives a definite numerical result once the partition and tags are fixed. It estimates accumulated signed area through a collection of rectangles. A finite sum should not be confused with the exact area under a curve or with a limit of sums. Whether sums approach a particular value as partitions become finer is a separate question.
For the construction itself, the essential information is local: every subinterval has one width, every subinterval has one tag, and the function must be defined at that tag. The sum may be positive, negative, or zero. A zero sum can occur because every contribution is zero, or because positive and negative contributions cancel.
Check Your Understanding
For each question, keep the tag, its function value, and its subinterval width paired.
- A partition has subintervals \([0,2]\) and \([2,5]\), with tags \(1\) and \(4\). If \(f(1)=3\) and \(f(4)=-2\), write the tagged sum without simplifying it.
- For the data in Question 1, calculate the tagged sum. What is the sign of each rectangle contribution?
- Can a tag be an endpoint of its subinterval? State the condition a tag must satisfy.
- If a tag has function value zero, what is that subinterval’s contribution to the sum?
- A rate is measured in metres per second and the subinterval width in seconds. What units does the corresponding term have?
