Why Sample at the Midpoints?
A rate describes how quickly a quantity changes. If the rate varies over an interval, knowing its value at just one point does not tell us the exact change across the whole interval. A practical estimate is to use a sampled rate as a representative rate for a short subinterval, multiply by the subinterval’s width, and add the estimated changes.
A midpoint sum follows this plan, with one specific choice: on each subinterval, sample the rate at the point halfway between its endpoints. The method works for equal-width and unequal-width partitions. The width of each subinterval must be matched with the rate sampled at that subinterval’s midpoint.
Definition of a Midpoint Sum
Partition an interval from \(a\) to \(b\) using increasing endpoints \(x_0,x_1,\ldots,x_n\), where \(x_0=a\) and \(x_n=b\). The \(i\)th subinterval is \([x_{i-1},x_i]\). Its width is \(x_i-x_{i-1}\), and its midpoint is halfway between its endpoints.
Here, \(m_i\) is the midpoint of the \(i\)th subinterval, and \(x_i-x_{i-1}\) is its width. Each term \(f(m_i)(x_i-x_{i-1})\) is one estimated contribution. The sum \(M\) is the total of those contributions.
If \(f\) is a rate, the product of a rate and a time width estimates the change in the quantity over that time interval. For example, a rate in metres per second multiplied by a time width in seconds gives an estimated change in metres. The units of the total are therefore the units of the accumulated quantity.
How to Build the Sum
Work through the subintervals one at a time. For each one, find its width and midpoint, evaluate the rate at that midpoint, and multiply the two values. Add the resulting products, retaining any negative signs. A midpoint sum can be written out term by term before being compressed into sigma notation.
Record their endpoints in increasing order, including any unequal widths.
For \([x_{i-1},x_i]\), use \(m_i=(x_{i-1}+x_i)/2\) and \(\Delta x_i=x_i-x_{i-1}\).
Find \(f(m_i)\), then calculate \(f(m_i)\Delta x_i\) for that same subinterval.
Sum all products, keeping their signs and stating the units when the quantities have units.
Worked Examples
Worked Example: A Formula on Unequal Subintervals
Estimate the accumulation of the rate \(f(t)=t^2+2\) from \(t=0\) to \(t=4\), using the partition with endpoints \(0,1,3,4\).
The subintervals are \([0,1]\), \([1,3]\), and \([3,4]\). Their widths are \(1-0=1\), \(3-1=2\), and \(4-3=1\).
Their midpoints are \((0+1)/2=0.5\), \((1+3)/2=2\), and \((3+4)/2=3.5\). Evaluate the rate at each midpoint:
Multiply each sampled rate by the width of its own subinterval. The midpoint sum is
Check: The three contributions are \(2.25\), \(12\), and \(14.25\); adding them gives \(28.5\). In particular, the middle rate is multiplied by width \(2\), not width \(1\). This estimate uses unequal widths and is expressed in rate-units multiplied by \(t\)-units.
Worked Example: Estimating Distance from Speed Samples
A vehicle’s speed is sampled at the midpoint of each of three time intervals. The intervals are \([0,2]\), \([2,5]\), and \([5,7]\) seconds. The sampled speeds are \(12\), \(10\), and \(14\) metres per second, respectively. Construct the midpoint estimate of the distance travelled.
The widths are \(2-0=2\) seconds, \(5-2=3\) seconds, and \(7-5=2\) seconds. Each sampled speed is already given at the corresponding interval’s midpoint, so multiply each speed by its matching width:
Check: The contributions are \(24\), \(30\), and \(28\) metres, since \((\text{metres per second})(\text{seconds})=\text{metres}\). Their sum is \(24+30+28=82\) metres. The three widths add to \(2+3+2=7\) seconds, matching the full time interval from \(0\) to \(7\).
Worked Example: Keeping a Negative Rate Contribution
Estimate the net accumulation of \(f(t)=1-t\) from \(t=0\) to \(t=3\), using the partition with endpoints \(0,1,3\).
The subintervals are \([0,1]\) and \([1,3]\), with widths \(1-0=1\) and \(3-1=2\). Their midpoints are \((0+1)/2=0.5\) and \((1+3)/2=2\). The sampled rates are
Now multiply each rate by its matching width and add the signed contributions:
Check: The first contribution is \(0.5\), and the second is \(-2\); together they give \(-1.5\). The negative sign is retained because the sampled rate on the second subinterval is negative. The result estimates net accumulation, not total change without regard to direction.
Why the Products Are Added
The construction follows from the meaning of a rate. If a rate were constant at value \(c\) throughout a subinterval of width \(\Delta x\), its change over that subinterval would be \(c\Delta x\). When a rate varies, a midpoint sum uses the sampled value \(f(m_i)\) as an estimate of a representative rate on the \(i\)th subinterval. The estimated change there is consequently \(f(m_i)\Delta x_i\).
This justifies the midpoint-sum formula. It does not guarantee that the estimate equals the actual accumulated change: within a subinterval, the rate may differ from its midpoint value. Shorter subintervals can give more detailed sampling, but the sum remains an estimate unless additional information establishes exactness.
A Common Midpoint-Sum Error
A frequent mistake is to calculate a midpoint correctly but multiply its rate by the wrong width. In the first worked example, the midpoint of \([1,3]\) is \(2\), and \(f(2)=6\). The interval’s width is \(3-1=2\), so its contribution is \(6(2)=12\).
An incorrect calculation might use \(6(1)=6\), perhaps because the other subintervals each have width \(1\). That substitutes a width belonging to a different interval. The correction is to pair each midpoint rate with the width of the subinterval whose midpoint it is:
Another error is to multiply by the midpoint itself rather than the subinterval width. The midpoint determines where to sample; the width determines how long that sampled rate is used in the estimate. These are different roles. For \([1,3]\), the sample location is \(2\), while the width is \(2\); they happen to have the same numerical value here, but that coincidence does not make them interchangeable. On \([0,1]\), for instance, the midpoint is \(0.5\), while the width is \(1\).
Interpreting and Checking a Midpoint Sum
Before accepting a result, check the construction rather than only the final arithmetic. Each subinterval should appear once; each midpoint should lie halfway between its endpoints; each width should be the right endpoint minus the left endpoint; and each sampled rate must be multiplied by the width from that same subinterval. If the rate can be negative, preserve its sign when forming the product.
Units provide a useful independent check. A rate in litres per minute multiplied by a width in minutes contributes litres. If the products do not have the units of the accumulated quantity, either the rate units or the interval-width units have been mishandled. For signed rates, the resulting sum represents an estimate of net change, consistent with the distinction between net change and total change.
Midpoint sums are useful whenever information about a changing rate is available at representative times or positions. They turn a collection of local rate samples into one estimate of overall accumulation. The essential discipline is to keep three items paired for every subinterval: its midpoint, its width, and the rate evaluated at that midpoint.
Check Your Understanding
For each question, identify the sample location and width separately before calculating a contribution.
- For the subinterval \([2,6]\), what are its midpoint and width?
- A rate is \(5\) units per second at the midpoint of a \(3\)-second subinterval. What contribution does this subinterval make to a midpoint sum, and what are its units?
- For \(f(t)=t+1\) on the partition with endpoints \(0,2,5\), write the midpoint sum without simplifying it.
- Why must a negative sampled rate produce a negative contribution when the subinterval width is positive?
- Describe one check that can reveal whether a rate has been multiplied by the wrong subinterval width.
