#!/usr/bin/env python
# coding: utf-8

# # Basic Numerical Integration: the Trapezoid Rule

# A simple illustration of the trapezoid rule for definite integration:
# 
# $$
# \int_{a}^{b} f(x)\, dx \approx \frac{1}{2} \sum_{k=1}^{N} \left( x_{k} - x_{k-1} \right) \left( f(x_{k}) + f(x_{k-1}) \right).
# $$
# <br>
# First, we define a simple function and sample it between 0 and 10 at 200 points

# In[1]:


get_ipython().run_line_magic('matplotlib', 'inline')
import numpy as np
import matplotlib.pyplot as plt


# In[2]:


def f(x):
    return (x-3)*(x-5)*(x-7)+85

x = np.linspace(0, 10, 200)
y = f(x)


# Choose a region to integrate over and take only a few points in that region

# In[3]:


a, b = 1, 8 # the left and right boundaries
N = 5 # the number of points
xint = np.linspace(a, b, N)
yint = f(xint)


# Plot both the function and the area below it in the trapezoid approximation

# In[4]:


plt.plot(x, y, lw=2)
plt.axis([0, 9, 0, 140])
plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)
plt.text(0.5 * (a + b), 30,r"$\int_a^b f(x)dx$", horizontalalignment='center', fontsize=20);


# Compute the integral both at high accuracy and with the trapezoid approximation

# In[5]:


from __future__ import print_function
from scipy.integrate import quad
integral, error = quad(f, a, b)
integral_trapezoid = sum( (xint[1:] - xint[:-1]) * (yint[1:] + yint[:-1]) ) / 2
print("The integral is:", integral, "+/-", error)
print("The trapezoid approximation with", len(xint), "points is:", integral_trapezoid)