scipy.integrate.trapezoid#
- scipy.integrate.trapezoid(y, x=None, dx=1.0, axis=- 1)#
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.
- Returns
- trapzfloat 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
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.
References
- 1
Wikipedia page: https://en.wikipedia.org/wiki/Trapezoidal_rule
- 2
Illustration image: https://en.wikipedia.org/wiki/File:Composite_trapezoidal_rule_illustration.png
Examples
>>> np.trapz([1,2,3]) 4.0 >>> np.trapz([1,2,3], x=[4,6,8]) 8.0 >>> np.trapz([1,2,3], dx=2) 8.0
Using a decreasing x corresponds to integrating in reverse:
>>> np.trapz([1,2,3], x=[8,6,4]) -8.0
More generally x is used to integrate along a parametric curve. This finds 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) >>> np.trapz(np.cos(theta), x=np.sin(theta)) 3.141571941375841
>>> a = np.arange(6).reshape(2, 3) >>> a array([[0, 1, 2], [3, 4, 5]]) >>> np.trapz(a, axis=0) array([1.5, 2.5, 3.5]) >>> np.trapz(a, axis=1) array([2., 8.])
