# Integration Exercise 1

## Imports

In [14]:
%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 [15]:
def trapz(f, a, b, N):
h = (b - a)/N
i = np.arange(1,N)
c = h*(0.5*f(a)+f(b)*0.5+f(a+i*h).sum())
return c

In [16]:
f = lambda x: x**2
g = lambda x: np.sin(x)

In [17]:
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 [18]: