Integration Exercise 1

Imports



In [1]:

    
%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 [5]:

    
integrate.quad?



In [10]:

    
def trapz(f, a, b, N):
    """Integrate the function f(x) over the range [a,b] with N points."""
    h=(b-a)/N
    integral=0
    while a<b:
        integral=f(a)*h+(f(a+h)-f(a))*h/2+integral
        a=a+h
    return integral



In [11]:

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



In [14]:

    
%%timeit
trapz(f, 0, 1, 1000)









    



1000 loops, best of 3: 943 µs per loop



In [13]:

    
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]:

    
integ1, err1 = integrate.quad(f,0.0,1.0)
print(integ1)
print(err1)
integ2, err2=integrate.quad(g,0.0,np.pi)
print(integ2)
print(err2)









    



0.33333333333333337
3.700743415417189e-15
2.0
2.220446049250313e-14



In [ ]:

    
assert True # leave this cell to grade the previous one