Integration with Python

GOALS:

  • Approximate area under curve
  • Find exact areas under curve
  • Visualize area under curve

RECALL!

\[\text{Area} = \int_a^b f(x) dx = \lim_{n \to \infty} \frac{b-a}{n} \sum_{i = 1}^n f(a + i*\text{width})\]
In [1]:

%matplotlib inline
import numpy as np
import sympy as sy
import matplotlib.pyplot as plt

Problem

Find area under \(f(x) = x^3 - 2x\) on \([0,2]\) with

  1. 4 rectangles
  2. \(n\) rectangles as \(n \to \infty\)
In [2]:

width = (2-0)/4

In [3]:

heights = [(i**3 - 2*i) for i in [.5, 1, 1.5, 2]]
heights

Out[3]:

[-0.875, -1, 0.375, 4]

In [4]:

areas = [i*width for i in heights]
areas

Out[4]:

[-0.4375, -0.5, 0.1875, 2.0]

In [5]:

sum(areas)

Out[5]:

1.25

In [6]:

x = np.linspace(0,2, 100)

In [7]:

x

Out[7]:

array([ 0.        ,  0.02020202,  0.04040404,  0.06060606,  0.08080808,
        0.1010101 ,  0.12121212,  0.14141414,  0.16161616,  0.18181818,
        0.2020202 ,  0.22222222,  0.24242424,  0.26262626,  0.28282828,
        0.3030303 ,  0.32323232,  0.34343434,  0.36363636,  0.38383838,
        0.4040404 ,  0.42424242,  0.44444444,  0.46464646,  0.48484848,
        0.50505051,  0.52525253,  0.54545455,  0.56565657,  0.58585859,
        0.60606061,  0.62626263,  0.64646465,  0.66666667,  0.68686869,
        0.70707071,  0.72727273,  0.74747475,  0.76767677,  0.78787879,
        0.80808081,  0.82828283,  0.84848485,  0.86868687,  0.88888889,
        0.90909091,  0.92929293,  0.94949495,  0.96969697,  0.98989899,
        1.01010101,  1.03030303,  1.05050505,  1.07070707,  1.09090909,
        1.11111111,  1.13131313,  1.15151515,  1.17171717,  1.19191919,
        1.21212121,  1.23232323,  1.25252525,  1.27272727,  1.29292929,
        1.31313131,  1.33333333,  1.35353535,  1.37373737,  1.39393939,
        1.41414141,  1.43434343,  1.45454545,  1.47474747,  1.49494949,
        1.51515152,  1.53535354,  1.55555556,  1.57575758,  1.5959596 ,
        1.61616162,  1.63636364,  1.65656566,  1.67676768,  1.6969697 ,
        1.71717172,  1.73737374,  1.75757576,  1.77777778,  1.7979798 ,
        1.81818182,  1.83838384,  1.85858586,  1.87878788,  1.8989899 ,
        1.91919192,  1.93939394,  1.95959596,  1.97979798,  2.        ])

In [8]:

def f(x):
    return x**3 - 2*x

In [10]:

f(2)

Out[10]:

4

In [11]:

plt.plot(x, f(x))
plt.fill_between(x, f(x), alpha = 0.3)
plt.title("Area under $f(x) = x^3 - 2x$")

Out[11]:

Text(0.5,1,'Area under $f(x) = x^3 - 2x$')
_images/07-Integrals-with-Python_11_1.png 
In [12]:

x = sy.Symbol('x')

In [13]:

sy.integrate(f(x), (x, 0, 2))

Out[13]:

0

In [14]:

sy.integrate(f(x))

Out[14]:

x**4/4 - x**2

In [15]:

(2**4)/4 - 2**2

Out[15]:

0.0