Integration Exercise 3

Imports



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%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from scipy import integrate

2d polar integration

The 2d polar integral of a scalar function $f(r, \theta)$ is defined as:

$$ I(r_{max}) = \int_0^{r_{max}} \int_0^{2\pi} f(r, \theta) r d\theta dr$$

Write a function integrate_polar(f, rmax) that performs this integral numerically using scipy.integrate.dblquad.



In [37]:

    
integrate.dblquad?



In [39]:

    
def integrate_polar(f, rmax):
    """Integrate the function f(r, theta) over r=[0,rmax], theta=[0,2*np.pi]"""
    F=lambda r,t:f(r,t)*r
    def t1(t):
        return 0*t
    def t2(t):
        return rmax
    I,e=integrate.dblquad(F,0,2*np.pi,t1,t2)
    return I



In [40]:

    
assert np.allclose(integrate_polar(lambda r,t: 1, 1.0), np.pi)
assert np.allclose(integrate_polar(lambda r, t: np.exp(-r)*(np.cos(t)**2), np.inf), np.pi)



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