Bk3_Ch18_02.py

###############
# Authored by Weisheng Jiang
# Book 3  |  From Basic Arithmetic to Machine Learning
# Published and copyrighted by Tsinghua University Press
# Beijing, China, 2022
###############

# Bk3_Ch18_02

import numpy as np
from sympy import *
from matplotlib import pyplot as plt 

x = Symbol('x')

f_x = exp(-x**2)

integrate(f_x,(x,-oo,oo))

integrate(exp(-x**2/2),(x,-oo,oo))

f_x_fcn = lambdify([x],f_x)

integral_f_x = integrate(f_x, x)
integral_f_x_fcn = lambdify([x],integral_f_x)

a = -0.5
b = 1

num = 201; # number of mesh grids
x_array     = np.linspace(-3,3,num)
x_a_b_array = np.linspace(a,b,num)

y_array     = f_x_fcn(x_array)
y_a_b_array = f_x_fcn(x_a_b_array)

fig, ax = plt.subplots()
ax.plot(x_array, y_array, 'b')
ax.axvline(x = a, color = 'r', linestyle = '-')
ax.axvline(x = b, color = 'r', linestyle = '-')
ax.axhline(y = 0, color = 'k', linestyle = '-')

ax.fill_between(x_a_b_array,
                y_a_b_array, 
                edgecolor = 'none', 
                facecolor = '#DBEEF3')

ax.set_xlim(x_array.min(), x_array.max())
ax.set_ylim(np.floor(y_array.min()), 
            np.ceil(y_array.max()))
plt.gca().spines['right'].set_visible(False)
plt.gca().spines['top'].set_visible(False)
ax.set_xlabel('x')
ax.set_ylabel('f(x)')

integral_a_b = integral_f_x_fcn(b) - integral_f_x_fcn(a)

integral_a_b_v2 = integrate(f_x, (x, a, b))
integral_a_b_v2 = float(integral_a_b_v2)

ax.set_title(r'$\int_a^b  f(x) = %0.3f$'%integral_a_b)

#%% plot integral function

t = Symbol('t')
integral_f_x_oo_t = integrate(f_x, (x,-oo,t))
integral_f_x_oo_t_fcn = lambdify([t],integral_f_x_oo_t)

t_array     = np.linspace(-3,3,num)

integral_f_x_oo_t_array = integral_f_x_oo_t_fcn(t_array)

fig, ax = plt.subplots()
ax.plot(t_array, integral_f_x_oo_t_array, 'b')
ax.axvline(x = a, color = 'r', linestyle = '-')
ax.axvline(x = b, color = 'r', linestyle = '-')

ax.axhline(y = integral_f_x_oo_t_fcn(a), 
           color = 'r', linestyle = '-')

ax.axhline(y = integral_f_x_oo_t_fcn(b), 
           color = 'r', linestyle = '-')

ax.set_xlim(t_array.min(), t_array.max())
ax.set_ylim(np.floor(integral_f_x_oo_t_array.min()), 
            np.ceil(integral_f_x_oo_t_array.max()))
plt.gca().spines['right'].set_visible(False)
plt.gca().spines['top'].set_visible(False)
ax.set_xlabel('x')
ax.set_ylabel('Integral, F(x)')
ax.grid(linestyle='--', linewidth=0.25, color=[0.75,0.75,0.75])

plt.show()