# scipy.integrate.tplquad

scipy.integrate.tplquad(func, a, b, gfun, hfun, qfun, rfun, args=(), epsabs=1.49e-08, epsrel=1.49e-08)

  Compute a triple (definite) integral.

  Return the triple integral of `func(z, y, x)` from `x = a..b`, `y = gfun(x)..hfun(x)`, and `z = qfun(x,y)..rfun(x,y)`.

  Parameters:
    func function

      A Python function or method of at least three variables in the order (z, y, x).

    a, b float

      The limits of integration in x: a < b

    gfun function or float

      The lower boundary curve in y which is a function taking a single floating point argument (x) and returning a floating point result or a float indicating a constant boundary curve.

    hfun function or float

      The upper boundary curve in y (same requirements as gfun).

    qfun function or float

      The lower boundary surface in z. It must be a function that takes two floats in the order (x, y) and returns a float or a float indicating a constant boundary surface.

    rfun function or float

      The upper boundary surface in z. (Same requirements as qfun.)

    args tuple, optional

      Extra arguments to pass to func.

    epsabs float, optional

      Absolute tolerance passed directly to the innermost 1-D quadrature integration. Default is 1.49e-8.

    epsrel float, optional

      Relative tolerance of the innermost 1-D integrals. Default is 1.49e-8.

  Returns:
    y float

      The resultant integral.

    abserr float

      An estimate of the error.

  See also

  `quad`

    Adaptive quadrature using QUADPACK

  `quadrature`

    Adaptive Gaussian quadrature

  `fixed_quad`

    Fixed-order Gaussian quadrature

  `dblquad`

    Double integrals

  `nquad`

    N-dimensional integrals

  `romb`

    Integrators for sampled data

  `simpson`

    Integrators for sampled data

  `ode`

    ODE integrators

  `odeint`

    ODE integrators

  `scipy.special`

    For coefficients and roots of orthogonal polynomials

  Notes

  For valid results, the integral must converge; behavior for divergent integrals is not guaranteed.

  Details of QUADPACK level routines

  `quad` calls routines from the FORTRAN library QUADPACK. This section provides details on the conditions for each routine to be called and a short description of each routine. For each level of integration, `qagse` is used for finite limits or `qagie` is used, if either limit (or both!) are infinite. The following provides a short description from [1] for each routine.

  qagse

    is an integrator based on globally adaptive interval subdivision in connection with extrapolation, which will eliminate the effects of integrand singularities of several types.

  qagie

    handles integration over infinite intervals. The infinite range is mapped onto a finite interval and subsequently the same strategy as in `QAGS` is applied.

  References

  [1]

  Piessens, Robert; de Doncker-Kapenga, Elise; Überhuber, Christoph W.; Kahaner, David (1983). QUADPACK: A subroutine package for automatic integration. Springer-Verlag. ISBN 978-3-540-12553-2.

  Examples

  Compute the triple integral of `x * y * z`, over `x` ranging from 1 to 2, `y` ranging from 2 to 3, `z` ranging from 0 to 1. That is, \(\int^{x=2}_{x=1} \int^{y=3}_{y=2} \int^{z=1}_{z=0} x y z \,dz \,dy \,dx\).

      >>> import numpy as np
      >>> from scipy import integrate
      >>> f = lambda z, y, x: x*y*z
      >>> integrate.tplquad(f, 1, 2, 2, 3, 0, 1)
      (1.8749999999999998, 3.3246447942574074e-14)

  Calculate \(\int^{x=1}_{x=0} \int^{y=1-2x}_{y=0} \int^{z=1-x-2y}_{z=0} x y z \,dz \,dy \,dx\). Note: qfun/rfun takes arguments in the order (x, y), even though `f` takes arguments in the order (z, y, x).

      >>> f = lambda z, y, x: x*y*z
      >>> integrate.tplquad(f, 0, 1, 0, lambda x: 1-2*x, 0, lambda x, y: 1-x-2*y)
      (0.05416666666666668, 2.1774196738157757e-14)

  Calculate \(\int^{x=1}_{x=0} \int^{y=1}_{y=0} \int^{z=1}_{z=0} a x y z \,dz \,dy \,dx\) for \(a=1, 3\).

      >>> f = lambda z, y, x, a: a*x*y*z
      >>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(1,))
          (0.125, 5.527033708952211e-15)
      >>> integrate.tplquad(f, 0, 1, 0, 1, 0, 1, args=(3,))
          (0.375, 1.6581101126856635e-14)

  Compute the three-dimensional Gaussian Integral, which is the integral of the Gaussian function \(f(x,y,z) = e^{-(x^{2} + y^{2} + z^{2})}\), over \((-\infty,+\infty)\). That is, compute the integral \(\iiint^{+\infty}_{-\infty} e^{-(x^{2} + y^{2} + z^{2})} \,dz \,dy\,dx\).

      >>> f = lambda x, y, z: np.exp(-(x ** 2 + y ** 2 + z ** 2))
      >>> integrate.tplquad(f, -np.inf, np.inf, -np.inf, np.inf, -np.inf, np.inf)
          (5.568327996830833, 4.4619078828029765e-08)