# 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:
    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.

  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.

  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.

  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

  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.])