Integration Exercise 1

Imports



In [20]:

    
%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 [21]:

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



In [22]:

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



In [23]:

    
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 [24]:

    
print(trapz(f, 0, 1, 1000))
print(integrate.quad(f, 0, 1)[0])
print('error: '+ str(integrate.quad(f, 0, 1)[1]))
print(trapz(g, 0, np.pi, 1000))
print(integrate.quad(g, 0, np.pi)[0])
print('error: '+ str(integrate.quad(g, 0, np.pi)[1]))









    



0.3333335
0.33333333333333337
error: 3.700743415417189e-15
1.99999835507
2.0
error: 2.220446049250313e-14



In [25]:

    
assert True # leave this cell to grade the previous one