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# scipy.integrate.simpson

scipy.integrate.simpson(y, *, x=None, dx=1.0, axis=-1, even=<object object>)

  Integrate y(x) using samples along the given axis and the composite Simpson’s rule. If x is None, spacing of dx is assumed.

  If there are an even number of samples, N, then there are an odd number of intervals (N-1), but Simpson’s rule requires an even number of intervals. The parameter ‘even’ controls how this is handled.

  Parameters:
    y array_like

      Array to be integrated.

    x array_like, optional

      If given, the points at which y is sampled.

    dx float, optional

      Spacing of integration points along axis of x. Only used when x is None. Default is 1.

    axis int, optional

      Axis along which to integrate. Default is the last axis.

    even{None, ‘simpson’, ‘avg’, ‘first’, ‘last’}, optional
      ‘avg’Average two results:

        1. use the first N-2 intervals with a trapezoidal rule on the last interval and

        2. use the last N-2 intervals with a trapezoidal rule on the first interval.

      ‘first’Use Simpson’s rule for the first N-2 intervals with

        a trapezoidal rule on the last interval.

      ‘last’Use Simpson’s rule for the last N-2 intervals with a

        trapezoidal rule on the first interval.

      None : equivalent to ‘simpson’ (default)

      ‘simpson’Use Simpson’s rule for the first N-2 intervals with the

        addition of a 3-point parabolic segment for the last interval using equations outlined by Cartwright [1]. If the axis to be integrated over only has two points then the integration falls back to a trapezoidal integration.

        New in version 1.11.0.

      Changed in version 1.11.0: The newly added ‘simpson’ option is now the default as it is more accurate in most situations.

      Deprecated since version 1.11.0: Parameter even is deprecated and will be removed in SciPy 1.14.0. After this time the behaviour for an even number of points will follow that of even=’simpson’.

  Returns:
    float

      The estimated integral computed with the composite Simpson’s rule.

  See also

  `quad`

    adaptive quadrature using QUADPACK

  `romberg`

    adaptive Romberg quadrature

  `quadrature`

    adaptive Gaussian quadrature

  `fixed_quad`

    fixed-order Gaussian quadrature

  `dblquad`

    double integrals

  `tplquad`

    triple integrals

  `romb`

    integrators for sampled data

  `cumulative_trapezoid`

    cumulative integration for sampled data

  `cumulative_simpson`

    cumulative integration using Simpson’s 1/3 rule

  `ode`

    ODE integrators

  `odeint`

    ODE integrators

  Notes

  For an odd number of samples that are equally spaced the result is exact if the function is a polynomial of order 3 or less. If the samples are not equally spaced, then the result is exact only if the function is a polynomial of order 2 or less.

  References

  [1]

  Cartwright, Kenneth V. Simpson’s Rule Cumulative Integration with MS Excel and Irregularly-spaced Data. Journal of Mathematical Sciences and Mathematics Education. 12 (2): 1-9

  Examples

      >>> from scipy import integrate
      >>> import numpy as np
      >>> x = np.arange(0, 10)
      >>> y = np.arange(0, 10)

      >>> integrate.simpson(y, x=x)
      40.5

      >>> y = np.power(x, 3)
      >>> integrate.simpson(y, x=x)
      1640.5
      >>> integrate.quad(lambda x: x**3, 0, 9)[0]
      1640.25

      >>> integrate.simpson(y, x=x, even='first')
      1644.5