What a Trapezoidal Sum Is For
Sometimes you want the exact value of an integral, but you don't have an equation for the function or you don't know an antiderivative. Instead, you may be given a table of values such as and asked to approximate the area under the curve.
A trapezoidal sum approximates the area under a curve by replacing the curved pieces with straight-line segments. Each pair of consecutive data points forms a trapezoid, and the total approximation is the sum of those trapezoid areas.
A trapezoidal sum is simply a collection of trapezoid areas placed side by side. The more intervals you use, the more closely the straight-line approximation usually follows the curve.
The Area of One Trapezoid
Recall that the area of a trapezoid is the average of its two parallel side lengths multiplied by the distance between them.
For a graph of a function, the two parallel sides are the function values at the endpoints of the interval. If the interval runs from to , the trapezoid has heights and .
This gives the basic trapezoidal approximation on a single interval.
The Trapezoidal Sum Formula
Suppose an interval is divided into equal subintervals. The width of each subinterval is
If the function values are , the trapezoidal sum is
Notice the pattern:
- The first and last function values are used once.
- Every interior function value is used twice.
- Everything is multiplied by .
For equally spaced data, remember the coefficient pattern as 1, 2, 2, 2, …, 2, 1. Then multiply the entire weighted sum by .
Step-by-Step Method
- Determine the interval. Identify the starting and ending values of .
- Count the subintervals. Determine how many trapezoids you are using.
- Find the width. If the points are equally spaced, calculate .
- Apply the trapezoidal formula. Use the first and last function values once and every interior value twice.
- Calculate and round appropriately. Unless the problem specifies otherwise, follow the rounding instructions given in the AP question.
Worked Example 1: Using a Table
The table gives values of . Approximate using a trapezoidal sum.
| 0 | 1 | 2 | 3 | 4 |
| 3 | 5 | 4 | 7 | 6 |
Why the Interior Values Are Doubled
The formula can look strange at first because the interior function values receive a coefficient of 2. There is a simple geometric reason.
Consider three consecutive points, , , and . The middle height belongs to two adjacent trapezoids: the trapezoid from to and the trapezoid from to .
Therefore, when all the trapezoid areas are added together, the interior value appears twice. The endpoint values belong to only one trapezoid each, so they appear once.
Worked Example 2: Unequal Function Values
Suppose the values of are given by the table below. Approximate .
| 1 | 2 | 3 | 4 | 5 | |
|---|---|---|---|---|---|
| 2 | 4 | 5 | 3 | 6 |
Trapezoidal Sums and the Definite Integral
A trapezoidal sum is an approximation to a definite integral. If is continuous on , then increasing the number of subintervals generally makes the trapezoidal approximation approach the exact integral:
On the AP exam, however, you will often be asked to calculate an approximation from a table rather than evaluate the exact integral.
A trapezoidal sum is an approximation. Do not replace the trapezoidal sum with an exact integral unless the problem specifically asks you to find the exact value and gives you enough information to do so.
Trapezoidal Sums vs. Left and Right Sums
Trapezoidal sums are closely related to left and right Riemann sums. In fact, the trapezoidal approximation can be viewed as the average of the left and right endpoint approximations when the subintervals have equal width.
This relationship is useful because it explains why the trapezoidal method often provides a better approximation than using only left or only right endpoints.
Worked Example 3: Connecting Left, Right, and Trapezoidal Sums
Suppose is represented by the following equally spaced values:
| 0 | 1 | 2 | 3 | |
|---|---|---|---|---|
| 2 | 5 | 4 | 8 |
Using a Calculator
On many AP Calculus questions, the data will be given in a table and the arithmetic can be done efficiently with a calculator. The important part is knowing what expression to enter.
For equally spaced data, you can enter the weighted sum directly:
Then multiply by the interval width. If your calculator has a list or statistical mode, you can also organize the function values into a list and calculate the weighted sum systematically.
Students often calculate correctly but forget to multiply by . The weighted sum alone is not the trapezoidal approximation.
When the Subintervals Are Not Equal
The compact formula assumes that all the subintervals have the same width. If the spacing between the -values is not constant, calculate each trapezoid separately.
Then add the individual trapezoid areas together.
Suppose the table gives:
| 0 | 1 | 3 | |
|---|---|---|---|
| 4 | 6 | 5 |
Common Mistakes
For equally spaced data, the interior values must be counted twice. The correct coefficient pattern is 1, 2, 2, …, 2, 1.
A trapezoid is not a rectangle. Its area is the average of the two parallel sides multiplied by the width. If you forget the , you will generally get an answer that is too large.
The width is , not simply the right endpoint. Check the interval and the number of subintervals before beginning the calculation.
Unless the function is linear on every subinterval, a trapezoidal sum is generally an approximation. Use an approximation symbol when appropriate.
The shortcut formula with coefficients works only when the -values are equally spaced. Otherwise, calculate each trapezoid separately.
AP Exam Strategy
Trapezoidal sums are often embedded in larger AP Calculus problems involving accumulation, tables, motion, or data analysis. When you see a table and the words "use a trapezoidal sum", immediately look for the following:
- What interval am I approximating?
- How many subintervals are there?
- Are the -values equally spaced?
- What is the width of each subinterval?
- Which values are endpoints and which are interior values?
- Does the problem ask for an approximation of a signed integral or a geometric area?
Before calculating anything, write the trapezoidal structure. For five equally spaced points, think . This makes it much harder to accidentally omit an interior value.
Practice Problems
The table gives values of . Use a trapezoidal sum to approximate .
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Use a trapezoidal sum with four equal subintervals to approximate if and the function values at are .
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For equally spaced data, explain why the interior function values receive a coefficient of 2 in the trapezoidal sum.
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The values of are , , and . Approximate using trapezoids.
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For equally spaced data, how is the trapezoidal sum related to the left and right Riemann sums?
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