From a Table to an Accumulation Estimate
A table of rate values can describe how quickly a quantity changes at different times or positions. To estimate the total change over an interval, pair a rate value with an interval width: rate multiplied by width gives an estimated change for that part of the interval. Add the estimated changes across all the parts.
The choice of rate value matters. A left sum uses the rate at the start of each subinterval; a right sum uses the rate at its end; and a midpoint sum uses the rate at its midpoint. A table-based calculation is possible only when the table supplies the values at the required sample locations.
Set Up the Partition and the Table
Suppose the interval from \(a\) to \(b\) is divided at the points \(x_0,x_1,\ldots,x_n\), where \(x_0=a\) and \(x_n=b\). The \(i\)th subinterval is \([x_{i-1},x_i]\), and its width is \(x_i-x_{i-1}\). The widths need not all be equal.
If the rate is \(r(x)\), a table may list \(x\)-values and their corresponding rates. For a left sum, choose the value at \(x_{i-1}\); for a right sum, choose the value at \(x_i\). For a midpoint sum, choose the value at the midpoint of that subinterval. The midpoint of \([x_{i-1},x_i]\) is the average of its endpoints.
Thus the three sums use different table entries, but each selected entry is multiplied by the width of the same subinterval. This matching is essential, especially when the widths are unequal.
Here \(L\), \(R\), and \(M\) denote the left-, right-, and midpoint rectangle sums. These are estimates of the accumulated change, not necessarily its exact value. If \(r\) is measured in units per minute and \(x\) in minutes, each product has units of the accumulated quantity.
Why the Products Are Added
On a short subinterval, treating the rate as approximately constant gives a simple estimate of the change: rate times elapsed width. If the chosen rate is \(r_i\) and the width is \(\Delta x_i\), the estimated change on that subinterval is \(r_i\Delta x_i\). The changes over consecutive subintervals combine by addition, so the total estimate is the sum of these products.
This reasoning also explains the units: multiplying “quantity per unit of \(x\)” by “units of \(x\)” leaves units of quantity. If rates can be negative, the products and their sum are signed; a negative rate contributes a negative change.
Worked Example: Three Estimates with Unequal Widths
Worked Example: Three Estimates with Unequal Widths
A flow rate \(r(t)\), in liters per minute, is recorded in this table. Estimate the total volume accumulated from \(t=0\) to \(t=6\) minutes using left, right, and midpoint sums on the partition \(0,1,3,6\).
| Time \(t\) (min) | 0 | 0.5 | 1 | 2 | 3 | 4.5 | 6 |
|---|---|---|---|---|---|---|---|
| Rate \(r(t)\) (L/min) | 2 | 2.5 | 3 | 4 | 5 | 6 | 7 |
Step 1: Find the widths. The subintervals are \([0,1]\), \([1,3]\), and \([3,6]\). Their widths are \(1-0=1\), \(3-1=2\), and \(6-3=3\) minutes.
Step 2: Select the entries and calculate the left sum. The left endpoints are \(0,1,3\), whose rates are \(2,3,5\) L/min. Match each rate, in order, to widths \(1,2,3\):
Step 3: Calculate the right sum. The right endpoints are \(1,3,6\), whose rates are \(3,5,7\) L/min:
Step 4: Calculate the midpoint sum. The midpoints are \((0+1)/2=0.5\), \((1+3)/2=2\), and \((3+6)/2=4.5\) minutes. The corresponding rates are \(2.5,4,6\) L/min:
Check: The three calculations use the correct sample locations for their respective methods, and all three use the matching widths \(1,2,3\). Each term has units \((\text{L/min})(\text{min})=\text{L}\), as required for a volume change.
Use the Table's Locations, Not Just Its Row Order
It is useful to mark the chosen sample locations before multiplying anything. For left and right sums, these locations are partition endpoints. For a midpoint sum, calculate each midpoint from the endpoints of its own subinterval, then look up the rate at that location.
A table might include values at many locations, not all of which are needed. Conversely, it might omit a location required for a particular sum. Do not treat the nearest listed value as though it were the required value. Unless an additional method or assumption justifies estimating a missing rate, the table alone does not determine that rectangle sum.
List the endpoints in order so that every subinterval is clear.
Subtract the left endpoint from the right endpoint of that same subinterval.
Use left endpoints, right endpoints, or the subinterval midpoints, as required.
Look up each rate and keep it paired with the width from its subinterval.
Include units, then check that every product represents a change in the accumulated quantity.
Worked Example: Signed Velocity Values
Worked Example: Signed Velocity Values
A particle's velocity is listed in meters per second. Use the partition \(0,2,4,6\) seconds to estimate its displacement with left, right, and midpoint sums.
| Time \(t\) (s) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Velocity \(v(t)\) (m/s) | 3 | 2 | 1 | 0 | -1 | -2 | -3 |
Step 1: Determine the widths. There are three subintervals: \([0,2]\), \([2,4]\), and \([4,6]\). Each has width \(2\) seconds.
Step 2: Use the left endpoints. The left-endpoint velocities are \(v(0)=3\), \(v(2)=1\), and \(v(4)=-1\), all in meters per second:
Step 3: Use the right endpoints. The right-endpoint velocities are \(v(2)=1\), \(v(4)=-1\), and \(v(6)=-3\):
Step 4: Use the midpoints. The midpoints are \(1,3,5\) seconds, where the velocities are \(2,0,-2\) m/s:
Check: Each term has units \((\text{m/s})(\text{s})=\text{m}\). Negative velocities give negative displacement contributions; they are not replaced by their absolute values. The calculations therefore estimate signed displacement, not total distance traveled.
A Common Width-Matching Error
With unequal intervals, it is tempting to calculate one representative width and use it for every term. That changes the problem. In the first example, the widths were \(1,2,3\) minutes, not a common width. For instance, the following left-sum calculation is wrong:
It incorrectly assigns width \(2\) to all three intervals. The correction is to pair the left-endpoint rates \(2,3,5\) with their own widths \(1,2,3\):
A related mistake is to choose the right sample but attach it to the next interval's width. Avoid this by writing one row for each subinterval, with its width and selected rate side by side, before evaluating the sum.
When the Table Does Not Contain the Needed Values
Suppose the partition is \(0,2,5\). A midpoint sum needs rates at \(1\) and \(3.5\), because those are the midpoints of \([0,2]\) and \([2,5]\). If the table contains values only at \(0,2,5\), it provides the endpoints needed for left and right sums, but it does not provide the required midpoint rates.
In that situation, do not label an estimate made from the nearest table entries as the midpoint sum. State which entries are available and which sample locations are missing. The same check applies to any proposed rectangle sum: the required entries are determined by the method and the partition, not by whichever table values happen to be easiest to use.
Check Your Understanding
For each question, pay attention to both the sample locations and the interval widths.
- A partition is \(0,2,5\). What are the widths of its two subintervals?
- For a right sum on the partition \(1,3,6\), which endpoints supply the rate values?
- For a midpoint sum on \([2,6]\), at what location must the table give a rate value?
- A rate is measured in gallons per minute and time in minutes. What are the units of a rate-times-width term?
- A table lists rates only at partition endpoints. Does it necessarily contain the entries needed for a midpoint sum? Explain.
