bundles / scipy 2.0.0.dev0+git20260907.41eeb59 / scipy / integrate / _quadrature / trapezoid

function

`scipy.integrate._quadrature:trapezoid`

source: build-install/usr/lib/python3.14/site-packages/scipy/integrate/_quadrature.py:22

Signature

def trapezoid(y, x = None, dx = 1.0, axis = -1)

Summary

Integrate along the given axis using the composite trapezoidal rule.

Extended Summary

If x is provided, the integration happens in sequence along its elements - they are not sorted.

Integrate y (x) along each 1d slice on the given axis, compute $\int y (x) d x$ . When x is specified, this integrates along the parametric curve $C$ given by $(x (t), y (t))$ , computing the line integral $\int_{C} y d x = \int y (t) \frac{d x}{d t} d t$ .

Parameters

y : array_like: Input array to integrate.
x : array_like, optional: The sample points corresponding to the y values. If x is None, the sample points are assumed to be evenly spaced dx apart. The default is None.
dx : scalar, optional: The spacing between sample points when x is None. The default is 1.
axis : int, optional: The axis along which to integrate. The default is the last axis.

Returns

trapezoid : float or ndarray: Definite integral of y = n-dimensional array as approximated along a single axis by the trapezoidal rule. If y is a 1-dimensional array, then the result is a float. If n is greater than 1, then the result is an n-1 dimensional array.

Notes

Image ^[2] illustrates trapezoidal rule -- y-axis locations of points will be taken from y array, by default x-axis distances between points will be 1.0, alternatively they can be provided with x array or with dx scalar. Return value will be equal to combined area under the red lines.

Examples

Use the trapezoidal rule on evenly spaced points:

import numpy as np
from scipy import integrate

✓

integrate.trapezoid([1, 2, 3])

✗

np.float64(4.0)

The spacing between sample points can be selected by either the ``x`` or ``dx`` arguments:

integrate.trapezoid([1, 2, 3], x=[4, 6, 8])
integrate.trapezoid([1, 2, 3], dx=2)

✗

np.float64(8.0)

Using a decreasing ``x`` corresponds to integrating in reverse:

integrate.trapezoid([1, 2, 3], x=[8, 6, 4])

✗

np.float64(-8.0)

More generally ``x`` is used to integrate along a parametric curve. We can estimate the integral :math:`\int_0^1 x^2 = 1/3` using:

x = np.linspace(0, 1, num=50)
y = x**2

✓

integrate.trapezoid(y, x)

✗

np.float64(0.33340274885464394)

Or estimate the area of a circle, noting we repeat the sample which closes the curve:

theta = np.linspace(0, 2 * np.pi, num=1000, endpoint=True)

✓

integrate.trapezoid(np.cos(theta), x=np.sin(theta))

✗

np.float64(3.141571941375841)

``trapezoid`` can be applied along a specified axis to do multiple computations in one call:

a = np.arange(6).reshape(2, 3)
a
integrate.trapezoid(a, axis=0)

✓

array([1.5, 2.5, 3.5])

integrate.trapezoid(a, axis=1)

✗

array([2., 8.])

Aliases

scipy.integrate.trapezoid