Trapezoidal rule (trapezoid) in SciPy

Introduction

The trapezoidal rule helps us find the area under a curve using simple shapes called trapezoids. It is a way to estimate integrals when we have data points.

You have a list of measurements over time and want to find the total accumulated value.

You want to estimate the area under a curve but only have points, not a formula.

You need a quick approximation of an integral in a simple way.

You want to calculate distance traveled from speed data sampled at intervals.

Syntax

SciPy

scipy.integrate.trapezoid(y, x=None, dx=1.0, axis=-1)

y is the array of values you want to integrate.

x is optional and represents the sample points corresponding to y. If not given, spacing dx is used.

Examples

Calculates area under points assuming equal spacing of 1.

SciPy

from scipy.integrate import trapezoid

# Simple example with equal spacing
y = [1, 2, 3, 4]
area = trapezoid(y)
print(area)

Calculates area using given x positions for each y.

SciPy

from scipy.integrate import trapezoid

# Example with custom x values
x = [0, 1, 2, 3]
y = [1, 2, 3, 4]
area = trapezoid(y, x=x)
print(area)

Calculates area assuming points are spaced by 0.5 units.

SciPy

from scipy.integrate import trapezoid

# Using dx for spacing
y = [1, 4, 9, 16]
area = trapezoid(y, dx=0.5)
print(area)

Sample Program

This program calculates the total distance traveled by integrating speed over time using the trapezoidal rule.

SciPy

from scipy.integrate import trapezoid
import numpy as np

# Suppose we measure speed (m/s) every second
speed = np.array([0, 3, 6, 9, 12])
time = np.array([0, 1, 2, 3, 4])

# Calculate distance traveled using trapezoidal rule
distance = trapezoid(speed, x=time)
print(f"Distance traveled: {distance} meters")

Important Notes

The trapezoidal rule is more accurate than just summing rectangles but less accurate than Simpson's rule.

Make sure your x values are sorted and represent the correct spacing.

If x is not provided, dx is used as the spacing between points.

Summary

The trapezoidal rule estimates area under curves using trapezoids between points.

Use scipy.integrate.trapezoid with your data points and optional x values.

This method is simple and useful for quick integral approximations from data.