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

Compute a double integral.

Return the double (definite) integral of func(y, x) from x = a..b and y = gfun(x)..hfun(x).


A Python function or method of at least two variables: y must be the first argument and x the second argument.

a, bfloat

The limits of integration in x: a < b

gfuncallable 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.

hfuncallable or float

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

argssequence, optional

Extra arguments to pass to func.

epsabsfloat, optional

Absolute tolerance passed directly to the inner 1-D quadrature integration. Default is 1.49e-8. dblquad tries to obtain an accuracy of abs(i-result) <= max(epsabs, epsrel*abs(i)) where i = inner integral of func(y, x) from gfun(x) to hfun(x), and result is the numerical approximation. See epsrel below.

epsrelfloat, optional

Relative tolerance of the inner 1-D integrals. Default is 1.49e-8. If epsabs <= 0, epsrel must be greater than both 5e-29 and 50 * (machine epsilon). See epsabs above.


The resultant integral.


An estimate of the error.

See also


single integral


triple integral


N-dimensional integrals


fixed-order Gaussian quadrature


integrator for sampled data


integrator for sampled data


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 double integral of x * y**2 over the box x ranging from 0 to 2 and y ranging from 0 to 1. That is, \(\int^{x=2}_{x=0} \int^{y=1}_{y=0} x y^2 \,dy \,dx\).

>>> import numpy as np
>>> from scipy import integrate
>>> f = lambda y, x: x*y**2
>>> integrate.dblquad(f, 0, 2, 0, 1)
    (0.6666666666666667, 7.401486830834377e-15)

Calculate \(\int^{x=\pi/4}_{x=0} \int^{y=\cos(x)}_{y=\sin(x)} 1 \,dy \,dx\).

>>> f = lambda y, x: 1
>>> integrate.dblquad(f, 0, np.pi/4, np.sin, np.cos)
    (0.41421356237309503, 1.1083280054755938e-14)

Calculate \(\int^{x=1}_{x=0} \int^{y=2-x}_{y=x} a x y \,dy \,dx\) for \(a=1, 3\).

>>> f = lambda y, x, a: a*x*y
>>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(1,))
    (0.33333333333333337, 5.551115123125783e-15)
>>> integrate.dblquad(f, 0, 1, lambda x: x, lambda x: 2-x, args=(3,))
    (0.9999999999999999, 1.6653345369377348e-14)

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

>>> f = lambda x, y: np.exp(-(x ** 2 + y ** 2))
>>> integrate.dblquad(f, -np.inf, np.inf, -np.inf, np.inf)
    (3.141592653589777, 2.5173086737433208e-08)