Why Use Midpoints?
A rectangle can approximate the signed area between a curve and the horizontal axis over an interval. Its width comes from the interval, and its height comes from a function value. The choice of where to evaluate the function determines the rectangle’s height.
A midpoint sum uses the function value at the center of each subinterval. This gives a consistent rule: divide the interval into equal pieces, find each piece’s midpoint, and use the function value there as the rectangle’s height. The resulting sum approximates the function’s accumulated signed change over the full interval.
Definition and Formula
Suppose a function \(f\) is defined on \([a,b]\), where \(a<b\). Divide this interval into \(n\) equal subintervals, with \(n\) a positive integer. Their common width is \(\Delta x=(b-a)/n\). The \(i\)th subinterval runs from \(a+(i-1)\Delta x\) to \(a+i\Delta x\), for \(i=1,\ldots,n\).
The midpoint of that subinterval is halfway between its endpoints. Denote it by \(x_i^*\). Its formula follows by averaging the endpoints:
The index \(i\) starts at \(1\) and ends at \(n\), so the formula includes one midpoint from every subinterval. The sum \(M_n\) is a numerical approximation, not automatically the exact accumulated value.
Why the Formula Works
On one subinterval, a rectangle with width \(\Delta x\) and height \(f(x_i^*)\) has signed area \(\Delta x\,f(x_i^*)\). The height is positive when the function value is positive and negative when it is negative. Adding the signed areas of all \(n\) rectangles gives the midpoint sum.
This construction approximates accumulation because each rectangle represents the function using one sampled height across a whole subinterval. If the function changes within a subinterval, the rectangle’s height will not match the curve at every point. The sum is therefore generally an approximation. Using the midpoint makes the sample location precise and places it inside its own subinterval.
This reasoning explains both parts of the method: the common width supplies the factor \(\Delta x\), and the midpoint evaluation supplies the height for each rectangle.
Worked Example: A Formula on an Interval
Worked Example: Calculate a Midpoint Sum from a Formula
Approximate the accumulated value of \(f(x)=x^2\) on \([0,4]\) using \(n=4\) equal subintervals.
Step 1: Find the width.
Step 2: Find the four midpoints. The subintervals are \([0,1]\), \([1,2]\), \([2,3]\), and \([3,4]\). Their midpoints are, respectively,
Step 3: Evaluate the function at each midpoint.
Step 4: Multiply each height by the width and add.
Check: The four rectangle contributions are \(1(0.25)=0.25\), \(1(2.25)=2.25\), \(1(6.25)=6.25\), and \(1(12.25)=12.25\). Their total is \(0.25+2.25+6.25+12.25=21\), matching the midpoint-sum calculation.
Calculating from Tabulated Midpoint Values
Sometimes the function is not given by a formula. Instead, a problem may provide its values at the subinterval midpoints. If those values and the common width are known, the same sum applies: multiply each listed function value by the width and add the results.
Check that the values used really correspond to the midpoints. Values at endpoints are not substitutes for midpoint values just because they are listed in a table. Also keep any units attached to the function values and the input variable: multiplying a rate by a time width, for example, gives an accumulated amount.
Worked Example: Use Rate Values at Midpoints
A rate \(r(t)\), measured in liters per minute, is recorded at the midpoints of four equal time intervals covering \(0\) to \(8\) minutes. The midpoint values are shown below. Use a midpoint sum to approximate the accumulated amount.
| Midpoint time \(t\) (minutes) | Rate \(r(t)\) (liters per minute) |
|---|---|
| 1 | 2.4 |
| 3 | 3.1 |
| 5 | 2.8 |
| 7 | 3.7 |
Step 1: Find the width of each interval. There are four equal intervals over eight minutes, so
Step 2: Confirm the sample times are midpoints. The four intervals are \([0,2]\), \([2,4]\), \([4,6]\), and \([6,8]\). Their midpoints are \(1\), \(3\), \(5\), and \(7\) minutes, as listed.
Step 3: Multiply each rate by the width and add.
The units are \((\text{liters per minute})(\text{minutes})=\text{liters}\), so the estimated accumulated amount is \(24\) liters.
Check: The four interval contributions are \(2(2.4)=4.8\), \(2(3.1)=6.2\), \(2(2.8)=5.6\), and \(2(3.7)=7.4\) liters. Adding them gives \(4.8+6.2+5.6+7.4=24\) liters, as calculated.
Worked Example: Midpoints with a Different Starting Value
Worked Example: Apply the Indexed Midpoint Formula
For \(f(x)=x^2+2\), calculate the midpoint sum on \([-2,4]\) using \(n=3\) equal subintervals.
Step 1: Calculate the width.
Step 2: Use the midpoint formula. Here \(a=-2\). For \(i=1,2,3\), the midpoint formula gives
Substituting each index:
Step 3: Evaluate \(f\) at those midpoints.
Step 4: Form the sum.
Check: The subintervals are \([-2,0]\), \([0,2]\), and \([2,4]\); their midpoints are \(-1\), \(1\), and \(3\), confirming the sample locations. The rectangle contributions are \(2(3)=6\), \(2(3)=6\), and \(2(11)=22\). Their sum is \(6+6+22=34\).
A Common Mistake: Using Endpoints Instead
A frequent error is to calculate the width correctly but then use an endpoint as a rectangle’s sample point. That produces a left or right endpoint sum, not a midpoint sum.
Worked Example: Correct an Endpoint-Selection Error
For \(f(x)=x^2\) on \([0,4]\) with \(n=4\), the width is \(1\). A mistaken calculation might use \(x=0,1,2,3\), obtaining
This is not the midpoint sum: \(0,1,2,3\) are the left endpoints of the four subintervals. The correction is to use \(0.5,1.5,2.5,3.5\), the actual midpoints. Thus
Check: Each corrected input is halfway between the endpoints of its subinterval, and the four corresponding rectangle contributions add to \(21\). The value \(14\) came from a different sample-point rule.
Reading and Interpreting the Result
A midpoint sum is built from signed rectangle areas. If all sampled function values are positive, all the contributions are positive. If some are negative, their contributions are negative and reduce the total. Therefore, the sum represents signed accumulation, not necessarily total geometric area.
When setting up a calculation, keep the sequence of choices clear:
For \(n\) equal pieces of \([a,b]\), compute \(\Delta x=(b-a)/n\).
Use the midpoint formula or average the two endpoints of each subinterval.
Evaluate the formula at every midpoint or read the corresponding midpoint values from the data.
Multiply each height by the common width, then add all contributions with their signs.
Keep exact values during the calculation when possible, and round only if the question asks for a rounded approximation. A numerical answer should also include units when the quantities have units.
Check Your Understanding
Use the midpoint-sum method to answer the following questions.
- On \([2,10]\), divide the interval into four equal subintervals. What is the common width, and what are the four midpoints?
- For \(f(x)=2x+1\) on \([0,4]\) with \(n=2\), calculate the midpoint sum. Show the width, midpoints, and sampled function values.
- A rate is measured in kilograms per hour and sampled at the midpoints of three equal intervals, each \(2\) hours wide. The rates are \(1.5\), \(2.0\), and \(2.5\) kilograms per hour. What accumulated amount does the midpoint sum estimate?
- Explain why using the left endpoint of every subinterval does not give a midpoint sum.
