Integration Exercise 1

Imports



In [31]:

    
%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 [45]:

    
def trapz(f, a, b, N):
    """Integrate the function f(x) over the range [a,b] with N points."""
    x = np.linspace(a,b,N+1)
    h = np.diff(x)[1]
    y = f(x)
    m = .5 * (y[1:(len(y)-1)] + y[2:])
    a = sum(h * m)
    return(a)



In [43]:

    
np.linspace(0,1,2)









    Out[43]:





array([ 0.,  1.])



In [44]:

    
trapz(f, 0, 1, 1000)









    Out[44]:





0.33333349983233068



In [24]:

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



In [25]:

    
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 [30]:

    
integrate.quad(f,0,1)[0] - trapz(f, 0, 1, 1000)
integrate.quad(g,0, np.pi)[0] - trapz(g,0,np.pi, 1000)









    Out[30]:





6.5665886923582661e-06



In [ ]:

    
assert True # leave this cell to grade the previous one