Integration Exercise 3

Imports



In [1]:

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

2d polar integration

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

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

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



In [74]:

    
def integrate_polar(f, rmax):
    """Integrate the function f(r, theta) over r=[0,rmax], theta=[0,2*np.pi]"""
    thmin=0
    thmax=2*np.pi
    
    
    rmin=lambda t: 0
    rm=lambda t: rmax
    
    f1=lambda r,t :f(r,t)*r  #started on this path from suggestion in Gitter chat
    
    I,e=dblquad(f1,thmin,thmax,rmin,rm)
    return (I)



In [76]:

    
integrate_polar(lambda r,t: 1, 1.0)









    Out[76]:





3.141592653589793



In [77]:

    
integrate_polar(lambda r, t: np.exp(-r)*(np.cos(t)**2),np.inf)









    Out[77]:





3.141592653589792



In [80]:

    
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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