QUAD

Overview

The QUAD function computes the definite integral of a real-valued function over a specified interval using adaptive quadrature. This is useful for numerically integrating functions where an analytical solution is difficult or impossible. The function to be integrated is provided as a table of x and y values (sampled points), and linear interpolation is used between points. The integration is performed using the scipy.integrate.quad method, which is based on the QUADPACK Fortran library and uses adaptive subdivision and extrapolation for high accuracy:

where $f(x)$ is the function defined by the input table, and $a$ and $b$ are the integration limits. For more details, see the scipy.integrate.quad documentation .

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

Usage

To use the function in Excel:

=QUAD(function_table, a, b, [epsabs], [epsrel])

• function_table (2D list, required): Table of two columns, where the first column is x values and the second is y values. Must have at least two rows, with x values strictly increasing.
• a (float, required): Lower limit of integration.
• b (float, required): Upper limit of integration.
• epsabs (float, optional, default=1.49e-8): Absolute error tolerance.
• epsrel (float, optional, default=1.49e-8): Relative error tolerance.

The function returns a single value (float): the estimated value of the definite integral, or an error message (string) if the input is invalid.

Examples

Example 1: Integrate $f(x) = x^2$ from 0 to 4

Inputs:

Excel formula:

=QUAD({0,0;1,1;2,4;3,9;4,16}, 0, 4)

Expected output:

This means the integral of $x^2$ from 0 to 4 is approximately 22.00.

Example 2: Integrate $f(x) = \sin(x)$ from 0 to $\pi$

Inputs:

Excel formula:

=QUAD({0,0;1.5708,1;3.1416,0}, 0, 3.1416)

Expected output:

This means the integral of $\sin(x)$ from 0 to $\pi$ is approximately 1.571.

Example 3: Integrate $f(x) = e^{-x}$ from 0 to 5

Inputs:

Excel formula:

=QUAD({0,1;2.5,0.0821;5,0.0067}, 0, 5)

Expected output:

This means the integral of $e^{-x}$ from 0 to 5 is approximately 1.464.

Example 4: Integrate $f(x) = 2x + 1$ from 1 to 3

Inputs:

Excel formula:

=QUAD({1,3;2,5;3,7}, 1, 3)

Expected output:

This means the integral of $2x+1$ from 1 to 3 is 10.00.

Python Code

from scipy.integrate import quad as scipy_quad
 
def quad(function_table, a, b, epsabs=1.49e-8, epsrel=1.49e-8):
    """
    Numerically integrate a function defined by a table of x, y values over [a, b] using adaptive quadrature.
 
    Args:
        function_table: 2D list, each row is [x, y].
        a: Lower limit of integration.
        b: Upper limit of integration.
        epsabs: Absolute error tolerance (default: 1.49e-8).
        epsrel: Relative error tolerance (default: 1.49e-8).
 
    Returns:
        The estimated value of the definite integral (float), or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate function_table
    if not isinstance(function_table, list) or len(function_table) < 2:
        return "Invalid input: function_table must be a 2D list with at least two rows."
    try:
        xs = [float(row[0]) for row in function_table]
        ys = [float(row[1]) for row in function_table]
    except Exception:
        return "Invalid input: function_table must contain numeric values."
    if any(x2 <= x1 for x1, x2 in zip(xs, xs[1:])):
        return "Invalid input: x values in function_table must be strictly increasing."
    def f(x):
        # Linear interpolation
        if x <= xs[0]:
            return ys[0]
        if x >= xs[-1]:
            return ys[-1]
        for i in range(1, len(xs)):
            if x < xs[i]:
                x0, x1 = xs[i-1], xs[i]
                y0, y1 = ys[i-1], ys[i]
                return y0 + (y1 - y0) * (x - x0) / (x1 - x0)
        return ys[-1]
    try:
        result, _ = scipy_quad(f, float(a), float(b), epsabs=float(epsabs), epsrel=float(epsrel))
    except Exception as e:
        return f"scipy.quad error: {e}"
    return round(result, 3)

Live Notebook

Edit this function in a live notebook .

Live Demo