Integration Exercise 2

Imports



In [33]:

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

Indefinite integrals

Here is a table of definite integrals. Many of these integrals has a number of parameters $a$, $b$, etc.

Find five of these integrals and perform the following steps:

Typeset the integral using LateX in a Markdown cell.
Define an integrand function that computes the value of the integrand.
Define an integral_approx funciton that uses scipy.integrate.quad to peform the integral.
Define an integral_exact function that computes the exact value of the integral.
Call and print the return value of integral_approx and integral_exact for one set of parameters.

Here is an example to show what your solutions should look like:

Example

Here is the integral I am performing:

$$ I = \int_0^\infty \frac{dx}{x^2 + a^2} = \frac{\pi}{2a} $$



In [34]:

    
def integrand(x, a):
    return 1.0/(x**2 + a**2)

def integral_approx(a):
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand, 0, np.inf, args=(a,))
    return I

def integral_exact(a):
    return 0.5*np.pi/a

print("Numerical: ", integral_approx(1.0))
print("Exact    : ", integral_exact(1.0))









    



Numerical:  1.5707963267948966
Exact    :  1.5707963267948966



In [35]:

    
assert True # leave this cell to grade the above integral

Integral 1

$$ I_1 = \int_{0}^{\infty} \frac{\sin ^{2}px}{x^{2}}\ dx=\frac{\pi p}{2} $$



In [36]:

    
def integrand1(x, p):
    return np.sin(p*x)**2/(x**2)

def integral_approx1(p):
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand1, 0, np.inf, args=(p,))
    return I

def integral_exact1(p):
    return p*np.pi/2

print("Numerical: ", integral_approx1(1.0))
print("Exact    : ", integral_exact1(1.0))









    



Numerical:  1.5708678849453777
Exact    :  1.5707963267948966



In [37]:

    
assert True # leave this cell to grade the above integral

Integral 2

$$ I_2 = \int_0^\infty \frac {x}{e^{x}-1}\ dx= \frac {\pi^2}{6} $$



In [38]:

    
def integrand2(x):
    return x/(np.exp(x)-1)

def integral_approx2():
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand2, 0, np.inf)
    return I

def integral_exact2():
    return np.pi**2/6

print("Numerical: ", integral_approx2())
print("Exact    : ", integral_exact2())









    



Numerical:  1.6449340668482264
Exact    :  1.6449340668482264



In [39]:

    
assert True # leave this cell to grade the above integral

Integral 3

$$ I_3 = \int_0^a \frac{dx}{\sqrt{a^{2}-x^{2}}}=\frac{\pi }{2} $$



In [40]:

    
def integrand3(x, a):
    return 1.0/((a**2-x**2 )**(.5))

def integral_approx3(a):
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand3, 0, a, args=(a,))
    return I

def integral_exact3(a):
    return np.pi/2

print("Numerical: ", integral_approx3(17))
print("Exact    : ", integral_exact3(17))









    



Numerical:  1.570796326793784
Exact    :  1.5707963267948966



In [41]:

    
assert True # leave this cell to grade the above integral

Integral 4

$$ I_4 =\int_0^\infty \frac{x \sin mx}{x^2+a^2}\ dx=\frac{\pi}{2}e^{-ma} $$



In [42]:

    
def integrand4(x, m, a):
    return (x*np.sin(m*x))/(x**2+a**2)

def integral_approx4(m, a):
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand4, 0, np.inf, args=(m,a,))
    return I

def integral_exact4(m, a):
    return (np.pi/2)*np.exp(-1*m*a)

print("Numerical: ", integral_approx4(.001,.001))
print("Exact    : ", integral_exact4(.001,.001))









    



Numerical:  2.209712068886707
Exact    :  1.570794756



In [43]:

    
assert True # leave this cell to grade the above integral

Integral 5

$$ I_5 = \int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi} $$



In [44]:

    
def integrand5(x):
    return (np.exp(-1*(x**2)))

def integral_approx5():
    # Use the args keyword argument to feed extra arguments to your integrand
    I, e = integrate.quad(integrand5, -1*np.inf, np.inf)
    return I

def integral_exact5():
    return np.pi**(1/2)

print("Numerical: ", integral_approx5())
print("Exact    : ", integral_exact5())









    



Numerical:  1.7724538509055159
Exact    :  1.7724538509055159



In [45]:

    
assert True # leave this cell to grade the above integral