scipy.integrate.

trapezoid#

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

Integrate along the given axis using the composite trapezoidal rule.

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 \(\int y(x) dx\). When x is specified, this integrates along the parametric curve, computing \(\int_t y(t) dt = \int_t y(t) \left.\frac{dx}{dt}\right|_{x=x(t)} dt\).

Parameters:
yarray_like

Input array to integrate.

xarray_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.

dxscalar, optional

The spacing between sample points when x is None. The default is 1.

axisint, optional

The axis along which to integrate. The default is the last axis.

Returns:
trapezoidfloat 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.

See also

cumulative_trapezoid, simpson, romb

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.

trapezoid has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

✅

n/a

CuPy

n/a

✅

PyTorch

✅

✅

JAX

⛔

⛔

Dask

✅

n/a

See Support for the array API standard for more information.

References

Examples

Use the trapezoidal rule on evenly spaced points:

>>> import numpy as np
>>> from scipy import integrate
>>> integrate.trapezoid([1, 2, 3])
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])
8.0
>>> integrate.trapezoid([1, 2, 3], dx=2)
8.0

Using a decreasing x corresponds to integrating in reverse:

>>> integrate.trapezoid([1, 2, 3], x=[8, 6, 4])
-8.0

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

>>> x = np.linspace(0, 1, num=50)
>>> y = x**2
>>> integrate.trapezoid(y, x)
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))
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
array([[0, 1, 2],
       [3, 4, 5]])
>>> integrate.trapezoid(a, axis=0)
array([1.5, 2.5, 3.5])
>>> integrate.trapezoid(a, axis=1)
array([2.,  8.])
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