# 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.

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.

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

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.])