Integration Exercise 2

Imports



In [1]:

    
%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:

$$ I_1 = \int_0^\infty \frac{dx}{x^2 + a^2} = \frac{\pi}{2a} $$



In [2]:

    
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 [3]:

    
assert True # leave this cell to grade the above integral

Integral 1

\begin{equation*} I_1 = \int_0^{a} {\sqrt{a^2 - x^2}} dx= \frac {\pi a^2}{4} \end{equation*}



In [24]:

    
import math



In [66]:

    
def integrand(x, a):
    return (math.sqrt(a**2 - x**2))

def integral_approx(a):
    I, err = integrate.quad(integrand, 0, a, args=(a,))
    return I

def integral_exact(a):
    return math.pi*a**2/4



In [69]:

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









    



Numerical:  0.7853981633974481
Exact:  0.7853981633974483



In [ ]:

    
assert True # leave this cell to grade the above integral

Integral 2

\begin{equation*} I_2 = \int_0^1 \frac {\ln{x}}{1+x} dx = - \frac {\pi^2}{12} \end{equation*}



In [61]:

    
def integrand(x):
    return math.log(x) / (1+x)

def integral_approx():
    I, err = integrate.quad(integrand, 0, 1)
    return I

def integral_exact():
    return -(math.pi**2)/12



In [62]:

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









    



Numerical:  -0.8224670334241143
Exact:  -0.8224670334241132



In [ ]:

    
assert True # leave this cell to grade the above integral

Integral 3

\begin{equation*} I_3 = \int_0^{\infty} \frac {x}{e^x -1} dx = \frac {\pi^2}{6} \end{equation*}



In [82]:

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

def integral_approx():
    I, err = integrate.quad(integrand, 0, np.inf)
    return I

def integral_exact():
    return math.pi**2/6



In [90]:

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









    



Numerical:  0.7853981633974483
Exact:  0.7853981633974483



In [91]:

    
assert True # leave this cell to grade the above integral

Integral 4

\begin{equation*} I_4 = \int_0^\infty e^{-ax^2} dx = \frac {1}{2} \sqrt{\frac{\pi}{a}} \end{equation*}



In [96]:

    
def integrand(x,a):
    return np.exp(-a*x**2)

def integral_approx(a):
    I, err = integrate.quad(integrand, 0, np.inf, args=(a,))
    return I

def integral_exact(a):
    return 1/2 * math.sqrt(math.pi/a)



In [97]:

    
print("Numerical: ", integral_approx(1))
print("Exact: ", integral_exact(1))









    



Numerical:  0.8862269254527579
Exact:  0.8862269254527579



In [ ]:

    
assert True # leave this cell to grade the above integral

Integral 5

\begin{equation*} I_5 = \int_0^{\frac{\pi}{2}} \sin^2{x} \: dx = \frac {\pi}{4} \end{equation*}



In [88]:

    
def integrand(x):
    return (np.sin(x)**2)

def integral_approx():
    I, err = integrate.quad(integrand, 0, np.pi/2)
    return I

def integral_exact():
    return math.pi/4



In [89]:

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









    



Numerical:  0.7853981633974483
Exact:  0.7853981633974483



In [ ]:

    
assert True # leave this cell to grade the above integral