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

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).


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

a, bfloat

The limits of integration in x: a < b

gfunfunction 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.

hfunfunction or float

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

qfunfunction 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.

rfunfunction or float

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

argstuple, optional

Extra arguments to pass to func.

epsabsfloat, optional

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

epsrelfloat, optional

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


The resultant integral.


An estimate of the error.

See also


Adaptive quadrature using QUADPACK


Adaptive Gaussian quadrature


Fixed-order Gaussian quadrature


Double integrals


N-dimensional integrals


Integrators for sampled data


Integrators for sampled data


ODE integrators


ODE integrators


For coefficients and roots of orthogonal polynomials


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.


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


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



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.


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)