Riemann Sums Using Unequal Subintervals (Data Tables)

Why Unequal Subintervals Matter

Most introductory Riemann-sum problems use equally spaced values of . That makes the width of every subinterval the same, so you can factor a single out of the sum.

But AP Calculus problems often give you a data table in which the -values are not evenly spaced. In that situation, there is no single common width. You must calculate the width of each individual subinterval.

The good news is that the underlying idea has not changed. A Riemann sum is still an approximation made by adding the areas of rectangles:

The only difference is that each rectangle may have a different width.

The Core Formula

Suppose a table gives values of at several points . If you use a left Riemann sum, the approximation is:

For a right Riemann sum:

And for a midpoint sum, if the midpoint values are supplied or can be determined:

The important feature is that every term has the form:

height × width. The height comes from the function value, while the width comes from the difference between consecutive -values.

How to Read an Unequally Spaced Data Table

Consider the following table:

0	4
1	7
3	5
6	8
10	6

Notice that the -values are not equally spaced:

That means the four subintervals have four different widths:

This is the main idea to recognize on the AP exam. Do not automatically use a single . First calculate the width of each interval.

Worked Example 1: Left Riemann Sum

The approximation is therefore 140 distance units.

Worked Example 2: Right Riemann Sum

So the right Riemann sum gives an estimated distance of 156 distance units.

Worked Example 3: Trapezoidal Sum with Unequal Intervals

The trapezoidal rule is especially useful with data tables because it uses both endpoints of each subinterval.

For one interval from to , the area of the trapezoid is:

For multiple unequal intervals, calculate each trapezoid separately:

Why You Cannot Use One Common

For equally spaced data, you may be used to writing a Riemann sum in a compact form such as:

That works because every interval has the same width.

For unequal intervals, however, there is no single common width. For example, if the -values are , then the widths are:

Using for every rectangle would incorrectly treat all four intervals as though they had the same width.

Riemann Sums as Accumulated Change

On the AP exam, a Riemann sum often appears in a word problem involving a rate such as velocity, flow rate, population growth, or water entering a tank.

If represents a rate of change, then multiplying the rate by a time interval gives an approximate amount of accumulated change:

When the intervals have different lengths, each interval contributes a different amount:

This is why the units matter. If velocity is measured in miles per hour and time is measured in hours, then:

The resulting Riemann sum has units of miles.

Reading the Table Efficiently

When you see a data-table Riemann-sum problem, organize the information mentally as three columns:

Interval	Width	Selected Height

Then calculate:

This simple organization prevents one of the most common AP Calculus errors: pairing the correct function value with the wrong interval width.

Common Mistakes

Practice Problems

The Key Idea to Remember

When subintervals are unequal, there is no shortcut involving one common width. Treat each interval separately:

Once you recognize that structure, data-table Riemann sums become mechanical. The AP exam is usually testing whether you can correctly connect each function value with the appropriate interval width.