TRAPEZOID

Overview

The TRAPEZOID function numerically integrates sampled data using the composite trapezoidal rule. This method estimates the area under a curve by approximating the region between each pair of points as a trapezoid and summing their areas. It is commonly used for integrating discrete data points or functions sampled at regular or irregular intervals. The y-axis locations of points will be taken from the y array, and the x-axis sample points must be provided with the x array. The return value will be equal to the combined area under the red lines in the illustration below.

The composite trapezoidal rule is given by:

\int_a^b f(x) dx \approx \sum_{i=1}^{n-1} \frac{y_{i} + y_{i+1}}{2} (x_{i+1} - x_{i})

where $y_i$ are the function values at points $x_i$ .

This implementation only supports a single column for both y and x. This is more restrictive than the underlying scipy.integrate.trapezoid function, which supports higher-dimensional arrays and more advanced use cases. For more details, see the scipy.integrate.trapezoid documentation and the Wikipedia page .

Image source: Wikipedia - Composite trapezoidal rule illustration

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=TRAPEZOID(y, x)

y (2D list, required): Single column table of values to integrate. Each row is a sample.
x (2D list, required): Single column table of sample points corresponding to y. Must have the same number of rows as y.

The function returns a float. Returns an error message (string) if the input is invalid.

Examples

Example 1: Integrate a non-linear vector

This example integrates the vector [1, 4, 9] (squares) with sample points [0, 1, 2].

Inputs:

y	x
1	0
4	1
9	2

Excel formula:


=TRAPEZOID({1;4;9},{0;1;2})

Expected output:

Result
9.0

This means the area under the curve defined by [1,4,9] is 9.0.

Example 2: Integrate with custom x values

This example integrates [1, 2, 3] with sample points [4, 6, 8].

Inputs:

y	x
1	4
2	6
3	8

Excel formula:


=TRAPEZOID({1;2;3},{4;6;8})

Expected output:

Result
8.0

This means the area under the curve with custom sample points is 8.0.

Python Code


from scipy.integrate import trapezoid as scipy_trapezoid
 
def trapezoid(y, x):
    """
    Integrate sampled data using the composite trapezoidal rule.
 
    Args:
        y: 2D list (single column) of values to integrate. Each row is a sample.
        x: 2D list (single column) of sample points. Must match number of rows in y.
 
    Returns:
        Float with the integral, or error message if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate y
    if not isinstance(y, list) or len(y) < 2:
        return "Invalid input: y must be a 2D list with at least two rows."
    try:
        if all(isinstance(row, list) and len(row) == 1 for row in y):
            y_arr = [float(row[0]) for row in y]
        else:
            return "Invalid input: y must be a single column 2D list."
    except Exception:
        return "Invalid input: y must be a 2D list of numbers."
    # Handle x
    if not isinstance(x, list) or len(x) != len(y):
        return "Invalid input: x must be a 2D list with the same number of rows as y."
    try:
        if all(isinstance(row, list) and len(row) == 1 for row in x):
            x_arr = [float(row[0]) for row in x]
        else:
            return "Invalid input: x must be a single column 2D list."
    except Exception:
        return "Invalid input: x must be a 2D list of numbers."
    # Try integration
    try:
        val = scipy_trapezoid(y_arr, x=x_arr)
        return float(round(val, 6))
    except Exception as e:
        return f"scipy.trapezoid error: {e}"

Live Notebook

Edit this function in a live notebook .