nquad Integration in SciPy

Question

I am trying to self-learn SciPy and evaluate the following quadruple integral using scipy.integrate.nquad:

$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1-x} (w+y) \:dz \:dy \:dx \:dw $$

I wrote the following code:

from scipy import integrate

def f(z, y, x, w):
    return w + y

def bounds(z):
    return [0, 1-z]



I = integrate.nquad(f, [bounds, [0,1], [0,1], [0,1]])

                    
print(I)

However, it gives the following error:

bounds() takes 1 positional argument but 3 were given

Any help to solve this quadruple integral is much appreciated.

Answer 1

Assuming it is the $w$ variable that has the range $[0,1-x]$

from  scipy.integrate import nquad

def func(w,x,y,z):
    return w+y

def range_z():
    return [0,1]

def range_y(z):
    return [0,1]

def range_x(y, z):
    return [0,1]

def range_w(x, y, z):
    return (0,1-x)

res=nquad(func, [range_w, range_x, range_y, range_z])
print(res)

Running it produces 0.41666666666666674 which matches the analytic answer 5/12.

Assuming it is the $z$ variable that has the range $[0,1-x]$,

from  scipy.integrate import nquad

def func(z,y,x,w):
    return w+y

def range_w():
    return [0,1]

def range_x(w):
    return [0,1]

def range_y(x,w):
    return [0,1]

def range_z(y,x,w):
    return (0,1-x)

res=nquad(func, [range_z, range_y, range_x, range_w])
print(res)

In this case the answer from the code is 0.5 which matches the analytic answer 1/2.