From a Table of Rates to an Accumulated Change
A table may give the rate at which a quantity changes at selected times. A free-response question can then ask how much the quantity changed over an interval. The rate values alone are not the accumulated change: they must be combined with the time widths over which the change occurs.
Because a table gives rates at only selected times, the exact accumulation is generally not known from the table alone. A numerical method uses those values to estimate the accumulation. The method named in the prompt matters: a left-endpoint sum and a trapezoidal sum use the same table differently, so they can produce different estimates.
Set Up the Quantities and Units
Suppose \(Q(t)\) is a quantity and \(r(t)=Q'(t)\) is its rate of change. If \(Q\) is measured in liters and \(t\) in minutes, then \(r(t)\) is measured in liters per minute. Accumulation over a time interval has units of liters, since
For times \(t_0<t_1<\cdots<t_n\), the width of the \(i\)th interval is \(t_{i+1}-t_i\). Do not assume all widths are equal unless the table shows that they are. A rate is measured at a time; a rate multiplied by an interval width estimates change over that interval.
When rates can be positive or negative, retain their signs in a signed numerical sum. A negative rate contributes a negative estimated change. The interpretation should say whether the estimated accumulation represents an increase or a decrease in the quantity.
Left-Endpoint and Trapezoidal Sums
For a left-endpoint sum, use the rate at the left end of each subinterval and multiply by that subinterval’s width. For a trapezoidal sum, average the rates at the two endpoints of each subinterval and multiply by its width.
If the prompt specifies a right-endpoint sum instead, use the right-endpoint rate on each subinterval: \(r(t_{i+1})(t_{i+1}-t_i)\). If it specifies a midpoint sum, the table must provide the rate at the midpoint of each subinterval. Endpoint data alone does not provide those midpoint rates, so it does not by itself support a midpoint sum.
| Method | Rate used on each subinterval | Subinterval contribution |
|---|---|---|
| Left endpoint | Value at the left endpoint | \(r(t_i)(t_{i+1}-t_i)\) |
| Right endpoint | Value at the right endpoint | \(r(t_{i+1})(t_{i+1}-t_i)\) |
| Trapezoidal | Average of endpoint values | \(\frac{r(t_i)+r(t_{i+1})}{2}(t_{i+1}-t_i)\) |
| Midpoint | Value at the subinterval midpoint | \(r(\frac{t_i+t_{i+1}}{2})(t_{i+1}-t_i)\) |
Why the Trapezoidal Formula Works
The trapezoidal rule treats the rate as changing linearly between consecutive table entries. On one subinterval from \(t_i\) to \(t_{i+1}\), write \(h=t_{i+1}-t_i\), \(r_i=r(t_i)\), and \(r_{i+1}=r(t_{i+1})\). Under linear interpolation, the rate at \(t\) is
Integrating this straight-line model over the subinterval gives the estimated accumulation there. Set \(u=t-t_i\), so \(u\) runs from \(0\) to \(h\). Then
This is the area of a trapezoid whose parallel sides have heights \(r_i\) and \(r_{i+1}\), and whose width is \(h\). Adding the estimate for each subinterval gives the trapezoidal sum. The result is an approximation based on the table; it is exact for the linear model on each subinterval, but the actual rate need not be linear between data points.
Worked Examples
Worked Example: Estimate Inflow with the Trapezoidal Rule
A tank’s inflow rate \(r(t)\), in liters per minute, is recorded at the following times. Use the trapezoidal rule to estimate the amount of water that entered the tank from \(t=0\) to \(t=12\) minutes.
The time widths are \(4-0=4\), \(8-4=4\), and \(12-8=4\) minutes. Apply the trapezoidal formula separately on each interval:
Add the three interval estimates: \(16+18+12=46\) liters. Thus the trapezoidal rule estimates that 46 liters of water entered the tank during the 12-minute interval. The units check because each average rate in liters per minute is multiplied by 4 minutes.
As a check, the combined formula gives \(\frac{3+5}{2}(4)+\frac{5+4}{2}(4)+\frac{4+2}{2}(4)=16+18+12=46\) liters, matching the interval-by-interval calculation.
Worked Example: Use a Left-Endpoint Sum with Unequal Intervals
A stream’s flow rate is measured in cubic meters per hour. Use a left-endpoint sum to estimate the amount of water that flowed past a point from hour \(1\) to hour \(10\).
The subinterval widths are not all the same:
Use the rate at the left endpoint of each interval:
The estimated accumulation is \(14+15+16=45\) cubic meters. Therefore, the left-endpoint estimate is that approximately 45 cubic meters of water flowed past the point from hour 1 through hour 10.
Check the width and units in each term: for example, \(7\text{ m}^3/\text{h}\) multiplied by \(2\text{ h}\) gives \(14\text{ m}^3\). The estimate uses the left-endpoint rates \(7,5,4\); using the final table value \(6\) in the last term would instead use a right endpoint and would not follow the requested method.
Worked Example: Apply the Trapezoidal Rule to Signed Rates
The rate of change of a stored material is given in kilograms per hour. Use the trapezoidal rule to estimate the accumulated change from hour \(0\) to hour \(4\).
The interval widths are \(1\), \(2\), and \(1\) hour. Keep the negative rate in the first interval:
The estimated accumulated change is \(-0.5+4+1.5=5\) kilograms. Thus the trapezoidal rule estimates a net increase of 5 kilograms in the stored material over the four hours. The negative first contribution indicates a decrease during that subinterval; it is not replaced by its positive magnitude.
Check: the three signed contributions sum to \((-0.5)+4+1.5=5\) kilograms, and every contribution has units \((\text{kg/h})(\text{h})=\text{kg}\).
A Reliable Process for a Free-Response Answer
Underline whether the prompt asks for a left, right, midpoint, or trapezoidal sum. Follow that method rather than choosing one based only on convenience.
Subtract consecutive time values. Use the actual width on each interval, including when the intervals are unequal.
Pair each rate with the correct width and use the endpoint or midpoint specified by the method.
State the estimated amount of change, attach the accumulated quantity’s units, and describe what the estimate means in the stated situation.
Common Mistake: Forgetting the Interval Width
Suppose a table lists an inflow rate of \(5\) liters per minute at the left endpoint of a 4-minute interval. A left-endpoint estimate for that interval is not \(5\) liters. That would confuse the rate with the amount accumulated.
The incorrect calculation is \(5\text{ L/min}=5\text{ L}\). The units do not agree: liters per minute is a rate, not an amount. The correction is to multiply by the interval width: \(5\text{ L/min}\times4\text{ min}=20\text{ L}\). For a full estimate, repeat this multiplication for each subinterval and then add the contributions.
Reporting the Estimate Clearly
A complete contextual conclusion should include three elements: the numerical estimate, the units, and the meaning in context. For example: “Using the trapezoidal rule, the inflow is estimated to be 46 liters over the 12-minute interval.” Avoid reporting only “46,” and do not label a rate in liters per minute as an accumulated amount in liters.
Also make clear that a numerical sum is an estimate from the available table values. Unless the prompt supplies additional information about how the rate behaves between times, the table does not determine the exact accumulation. The indicated method provides the requested approximation.
Check Your Understanding
For each question, identify the method and pay attention to the units and time widths.
- A rate is given in gallons per minute at times \(0, 2,\) and \(5\) minutes. What are the two subinterval widths, and why must they be treated separately in a left-endpoint sum?
- For rates \(4\) and \(8\) meters per second at the endpoints of a 3-second interval, what is the trapezoidal estimate of the accumulation on that interval?
- A table provides only endpoint rates for each interval. Can those values alone be used for a midpoint sum? Explain.
- If a rate is negative during one interval, should its contribution to a signed trapezoidal sum be changed to a positive value? Explain briefly.
- What three pieces of information belong in a clear contextual conclusion after calculating an accumulation estimate?
