Why Midpoint Locations Matter
A midpoint sum uses one sample from the center of each subinterval. To build the sum correctly, we need to locate those centers precisely. This is especially important when the quantity being sampled is a rate: the rate value at each midpoint is multiplied by the subinterval's width to estimate the accumulated change over that interval.
The central task is to turn the interval endpoints and the number of subintervals into a formula for every sample location. Once that formula is established, it can be substituted directly into a rate function.
Equal-Width Subintervals and Their Midpoints
Let the interval be \([a,b]\), where \(a<b\), and divide it into \(n\) equal-width subintervals, with \(n\in\mathbb{N}\). The common width is the total interval length divided by the number of subintervals:
Number the subintervals from left to right using \(i=1,2,\ldots,n\). The endpoints of the \(i\)th subinterval are
Its midpoint is halfway between these two endpoints. Averaging them gives the midpoint location \(m_i\):
The index starts at \(1\), because the first subinterval runs from \(x_0=a\) to \(x_1=a+\Delta x\). Its midpoint is \(a+\Delta x/2\), not \(a-\Delta x/2\). The final midpoint is \(m_n=b-\Delta x/2\), so each sample lies half a subinterval-width inside the interval's endpoints.
Why the Midpoint Formula Works
The formula follows from the definition of a midpoint as the average of two endpoints. Here is the derivation in full, starting from the indexed endpoints.
The \(i\)th subinterval begins at \(x_{i-1}=a+(i-1)\Delta x\) and ends at \(x_i=a+i\Delta x\).
By the midpoint definition, \(m_i=(x_{i-1}+x_i)/2\).
Substitution gives \(m_i=[a+(i-1)\Delta x+a+i\Delta x]/2=a+(i-\frac12)\Delta x\).
Replacing \(\Delta x\) by \((b-a)/n\) gives \(m_i=a+(i-\frac12)(b-a)/n\).
This derivation also shows why the half-width appears: the midpoint is one-half of a subinterval-width beyond its left endpoint. The equivalent expression \(a+(2i-1)(b-a)/(2n)\) is useful when a single fraction is convenient.
From Midpoint Locations to a Rate Sum
Suppose a rate is described by a function \(r\) on \([a,b]\). On subinterval \(i\), use the rate value \(r(m_i)\) at the midpoint and multiply it by the width \(\Delta x\). Adding the contributions gives the midpoint sum
The first expression emphasizes the rate sampled at each midpoint. The second writes every sample location directly in terms of \(a\), \(b\), \(n\), and its index \(i\). This sum is an estimate of accumulated change over the interval. If the rate has units of quantity per unit of \(x\), multiplying by a width in units of \(x\) gives units of quantity.
Worked Example: A Quadratic Rate on Four Subintervals
A rate \(r(t)=t^2+1\) is measured in gallons per hour, with \(t\) in hours. Write and evaluate the midpoint sum on \([0,4]\) using \(n=4\) equal subintervals.
First calculate the width:
The midpoint formula is \(m_i=0+(i-\frac12)(1)\). For \(i=1,2,3,4\), the sample locations are \(0.5,\ 1.5,\ 2.5,\ 3.5\) hours. They are the centers of \([0,1]\), \([1,2]\), \([2,3]\), and \([3,4]\), respectively.
Evaluate the rate at each location:
Each rate value is in gallons per hour. The midpoint sum is therefore
Check: The midpoint locations are each \(0.5\) hour from the ends of their respective one-hour intervals, and the four rate values add to \(25\) gallons per hour. Multiplication by the one-hour width gives \(25\) gallons, as stated.
Using the Index Formula Efficiently
When \(a\) is not zero, the index formula helps avoid restarting the count at the wrong place. It can be used in two stages: first calculate the list of midpoint locations, then substitute those locations into the rate formula. Keeping those stages visible makes it easier to catch errors before evaluating a sum.
Worked Example: A Rate on an Interval That Starts Above Zero
Let \(r(t)=2t+4\) liters per hour for \(2\leq t\leq 8\). Find the midpoint sum using \(n=3\) equal subintervals.
The width is
Using \(m_i=a+(i-\frac12)\Delta t\), with \(a=2\), gives
Thus \(m_1=3\), \(m_2=5\), and \(m_3=7\) hours. These are the centers of \([2,4]\), \([4,6]\), and \([6,8]\). The sampled rates are
Therefore, the midpoint sum is
Check: Each sample is two hours from the previous sample, and the first and last samples are one hour inside the interval's endpoints. The three rate values total \(42\) liters per hour; multiplying by the two-hour width gives \(84\) liters.
Writing a Sum Before Evaluating It
Sometimes the important result is the correctly constructed sum, not a decimal or numerical total. A useful check is to compare the formula inside the rate function with the midpoint formula: every sample should use the same interval and subinterval count, with only the index changing.
Worked Example: Constructing and Evaluating a Sum from the Formula
Suppose \(q(t)=t^2-2t+5\) liters per minute on \([1,5]\). Construct the midpoint sum for \(n=2\) subintervals and evaluate it.
The width is
The two midpoint locations follow from \(m_i=1+(i-\frac12)2\):
Substitute these values into the rate formula:
The sum, written first in indexed form, is
Evaluating it with units gives
Check: The intervals are \([1,3]\) and \([3,5]\), whose midpoints are \(2\) and \(4\). The rate values add to \(18\) liters per minute, and multiplying by the two-minute width gives \(36\) liters.
A Common Indexing Mistake
A frequent error is to use the right endpoint formula where the midpoint formula is needed. For example, on \([2,10]\) with \(n=4\), the width is \(\Delta x=(10-2)/4=2\). The incorrect expression \(2+i\Delta x\) gives \(4,6,8,10\). Those are the right endpoints, not the midpoints.
The correct midpoint locations are
so they are \(3,5,7,9\). Checking the first subinterval exposes the error immediately: its endpoints are \(2\) and \(4\), whose average is \((2+4)/2=3\), not \(4\). The half-width adjustment is essential.
A Practical Check Before Summing
Before evaluating a midpoint sum, verify these three features:
- The width is \((b-a)/n\), with \(n\) a positive integer.
- The first and last midpoint locations are \(a+\Delta x/2\) and \(b-\Delta x/2\).
- Each sampled rate is multiplied by one subinterval-width, and the resulting units are units of accumulated quantity.
These checks do not replace the calculation, but they catch common mistakes in endpoint choice, indexing, and units. A correctly written sum also makes clear what is being added: one midpoint rate contribution from each of the \(n\) equal-width subintervals.
Check Your Understanding
Use the midpoint formula \(m_i=a+(i-\frac12)(b-a)/n\) where needed.
- Divide \([3,15]\) into four equal subintervals. What is the width, and what are the four midpoint locations?
- For \([0,6]\) with \(n=3\), write the midpoint location \(m_i\) as a formula involving \(i\), then list the locations.
- A rate is \(r(t)=t+2\) units per hour on \([1,5]\). Write the midpoint sum for \(n=2\), including the units on the accumulated result.
- For \([2,10]\) with \(n=4\), explain why \(2+2i\) gives right endpoints rather than midpoint locations.
- If a rate has units of liters per minute and the interval width is measured in minutes, what units does each rate-times-width contribution have?
