Integration Exercise 1

Imports



In [4]:

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

Trapezoidal rule

The trapezoidal rule generates a numerical approximation to the 1d integral:

$$ I(a,b) = \int_a^b f(x) dx $$

by dividing the interval $[a,b]$ into $N$ subdivisions of length $h$:

$$ h = (b-a)/N $$

Note that this means the function will be evaluated at $N+1$ points on $[a,b]$. The main idea of the trapezoidal rule is that the function is approximated by a straight line between each of these points.

Write a function trapz(f, a, b, N) that performs trapezoidal rule on the function f over the interval $[a,b]$ with N subdivisions (N+1 points).



In [14]:

    
def trapz(f, a, b, N):
    """Integrate the function f(x) over the range [a,b] with N points."""
    h=(b-a)/N
    k=np.arange(1,N)
    I=h*(0.5*f(a)+0.5*f(b)+sum(f(a+k*h)))
    return I



In [15]:

    
f = lambda x: x**2
g = lambda x: np.sin(x)



In [18]:

    
I = trapz(f, 0, 1, 1000)
assert np.allclose(I, 0.33333349999999995)
J = trapz(g, 0, np.pi, 1000)
assert np.allclose(J, 1.9999983550656628)

Now use scipy.integrate.quad to integrate the f and g functions and see how the result compares with your trapz function. Print the results and errors.



In [19]:

    
I,e=integrate.quad(f,0,1)
print('Result:',I,"error:",e)









    



Result: 0.33333333333333337 error: 3.700743415417189e-15



In [20]:

    
I,e=integrate.quad(g,0,np.pi)
print('Result:',I,"error:",e)









    



Result: 2.0 error: 2.220446049250313e-14

The results of the trapezoidal method and the integrate function are very close with a difference ~$10^{-7}$. This shows that the integrate function works.



In [ ]:

    
assert True # leave this cell to grade the previous one