How to do trapezoidal sum with table is a practical question when you need to estimate the area under a curve from a set of recorded values instead of a continuous formula. That said, in numerical integration, the trapezoidal sum approximates a definite integral by dividing the region into small trapezoids, each formed by two adjacent data points in the table. This method is especially useful in physics, engineering, statistics, and calculus classes because it turns a table of x and y values into a single numerical estimate of accumulated change, total distance, total cost, or total flow.
Why Use a Trapezoidal Sum from a Table?
Many real-world problems do not give you a neat function such as f(x) = x². Instead, they give you measurements taken at specific times or positions. Even so, for example, a scientist may record temperature every hour, a car may have speed readings every ten seconds, or a company may track revenue at the end of each month. In these cases, you cannot integrate a formula directly, but you can still estimate the total using the data you have Worth knowing..
The trapezoidal sum works by treating each pair of adjacent points as the top and bottom of a trapezoid. The height of each trapezoid comes from the y values, and the width comes from the difference between consecutive x values. When you add the areas of all the trapezoids, you get an approximation of the total area under the curve.
called the Trapezoidal Rule. It strikes a practical balance: it is more accurate than using rectangles (left or right Riemann sums) because the slanted top edge follows the general trend of the data, yet it remains simple enough to calculate by hand or with a basic spreadsheet.
The Core Formula
Suppose your table provides $n$ data points $(x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)$ where the $x$-values are in increasing order. The trapezoidal approximation for the integral $\int_{x_0}^{x_n} f(x) , dx$ is the sum of the areas of the $n-1$ trapezoids formed between each consecutive pair of points Most people skip this — try not to. Nothing fancy..
For a single subinterval $[x_{i-1}, x_i]$, the area of the trapezoid is: $ \text{Area}i = \frac{y{i-1} + y_i}{2} \cdot (x_i - x_{i-1}) $
The Total Trapezoidal Sum is therefore: $ T = \sum_{i=1}^{n} \frac{y_{i-1} + y_i}{2} \Delta x_i $ where $\Delta x_i = x_i - x_{i-1}$.
Special Case (Equal Spacing): If the $x$-values are evenly spaced (e.g., time intervals of 1 hour), then $\Delta x_i = h$ is constant. The formula simplifies to the familiar "average of the ends plus twice the middles" pattern: $ T = \frac{h}{2} \left[ y_0 + 2y_1 + 2y_2 + \dots + 2y_{n-1} + y_n \right] $
Step-by-Step Procedure
- Verify Order: Ensure the table is sorted by the independent variable ($x$ or $t$) ascending.
- Calculate Widths: For each row $i$ (starting from the second row), compute $\Delta x_i = x_i - x_{i-1}$.
- Calculate Average Heights: For each interval, compute the average of the two $y$-values: $\frac{y_{i-1} + y_i}{2}$.
- Compute Partial Areas: Multiply the average height by the width for that interval: $\text{Area}i = \left(\frac{y{i-1} + y_i}{2}\right) \Delta x_i$.
- Sum: Add all partial areas together to get the final estimate.
- Attach Units: The units of the result are (units of $y$) $\times$ (units of $x$).
Worked Example: Uneven Time Intervals
A drone’s velocity (m/s) is recorded at uneven time intervals (seconds). Estimate the total distance traveled from $t=0$ to $t=10$ Worth knowing..
| $t$ (s) | 0 | 2 | 5 | 7 | 10 |
|---|---|---|---|---|---|
| $v(t)$ (m/s) | 10 | 14 | 18 | 15 | 12 |
Step 1: Calculate widths ($\Delta t$)
- $[0, 2]$: $\Delta t = 2$
- $[2, 5]$: $\Delta t = 3$
- $[5, 7]$: $\Delta t = 2$
- $[7, 10]$: $\Delta t = 3$
Step 2: Calculate average velocities & areas
- $t=0 \to 2$: Avg $v = \frac{10+14}{2} = 12$; Area $= 12 \times 2 = \mathbf{24 \text{ m}}$
- $t=2 \to 5$: Avg $v = \frac{14+18}{2} = 16$; Area $= 16 \times 3 = \mathbf{48 \text{ m}}$
- $t=5 \to 7$: Avg $v = \frac{18+15}{2} = 16.5$; Area $= 16.5 \times 2 = \mathbf{33 \text{ m}}$
- $t=7 \to 10$: Avg $v = \frac{15+12}{2} = 13.5$; Area $= 13.5 \times 3 = \mathbf{40.5 \text{ m}}$
Step 3: Total Distance $ 24 + 48 + 33 + 40.5 = \mathbf{145.5 \text{ meters}} $
Accuracy, Error, and Refinement
The trapezoidal sum is exact for linear functions because the top of the trapezoid matches the function perfectly. For curves, the error depends on the concavity:
- **Concave Up ($f''
