# 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:

1. Typeset the integral using LateX in a Markdown cell.
2. Define an integrand function that computes the value of the integrand.
3. Define an integral_approx funciton that uses scipy.integrate.quad to peform the integral.
4. Define an integral_exact function that computes the exact value of the integral.
5. 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]:

#I worked with James Amarel on this assignement
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 [ ]:

assert True # leave this cell to grade the above integral



### Integral 1

$$I_1 = \int_0^ a \sqrt{a^2 - x^2}dx = \frac{\pi a^2}{4}$$


In [8]:

#raise NotImplementedError()
def integrand(x, a):
return np.sqrt(a**2 - x**2)

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

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

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

$$I_2 = \int_0^\frac{\pi}{2} \sin^2(x)dx = \frac{\pi}{4}$$


In [13]:

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

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

def integral_exact():
return np.pi/4

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



### Integral 3

$$I_3 = \int_0^{2\pi} \frac{dx}{a+b\sin(x)} = \frac{2\pi}{\sqrt{a^2-b^2}}$$



In [34]:

#raise NotImplementedError()
def integrand(x,a,b):
return 1.0/(a+b*np.sin(x))

def integral_approx(a,b):
I, e = integrate.quad(integrand, 0, 2*np.pi, args=(a,b))
return I

def integral_exact(a,b):
return (2*np.pi)/np.sqrt(a**2-b**2)

print("Numerical: ", integral_approx(10.0,1.0))
print("Exact    : ", integral_exact(10.0,1.0))




Numerical:  0.6314838833996557
Exact    :  0.6314838834




In [ ]:

assert True # leave this cell to grade the above integral



### Integral 4

$$I_4 = \int_0^\infty e^{-ax^2} dx = \frac{1}{2}\sqrt{\frac{\pi}{2a}}$$


In [41]:

#raise NotImplementedError()
def integrand(x, a):
return np.e**(-1.0*a*(x**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.sqrt(np.pi/(a))

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




Numerical:  0.886226925452758
Exact    :  0.886226925453




In [ ]:

assert True # leave this cell to grade the above integral



### Integral 5

$$I_5 = \int_{-\infty}^\infty \frac{1}{\cosh x} dx = \pi$$


In [43]:

#raise NotImplementedError()
def integrand(x):
return 1.0/np.cosh(x)

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

def integral_exact():
return np.pi

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




Numerical:  3.1415926535897936
Exact    :  3.141592653589793




In [ ]:

assert True # leave this cell to grade the above integral