How to find Definite Integral using Python ?

Definite integrals are the extension after indefinite integrals, definite integrals have limits [a, b]. It gives the area of a curve bounded between given limits.

\int_{a}^b F(x)dx

It denotes the area of curve F(x) bounded between a and b, where a is the lower limit and b is the upper limit.

In this article, we will discuss how we can solve definite integrals in python, and would also visualize the area between them using matplotlib. We would also use the NumPy module for defining the range of the variable we are integrating. Let's Begin with installing the modules.

Module needed:

• matplotlib: We would use this to visualize our area under the graph formed by a definite integral.
• numpy: Helper library to define ranges of definite integrals.
• sympy: Library to calculate the numerical solution of the integral easily.

Approach

For calculating area under curve

• Import module
• Declare function
• Integrate.

Syntax : 

sympy.integrate(expression, reference variable)

For plotting 

• Import module
• Define a function
• Define a variable
• Draw the curve
• Fill the color under it using some condition.
• Display plot

Given below is the implementation for the same.

The area between a curve and standard axis

Example 1 :

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


def f(x):
    return x**2

x = sy.Symbol("x")
print(sy.integrate(f(x), (x, 0, 2)))

Output:

8/3

Example 2:

import matplotlib.pyplot as plt
import numpy as np


def f(x):
    return x**2

x = np.linspace(0, 2, 1000)
plt.plot(x, f(x))
plt.axhline(color="black")
plt.fill_between(x, f(x), where=[(x > 0) and (x < 2) for x in x])
plt.show()

Output:

The area between two curves

Example 1:

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


def f(x):
    return x**2


def g(x):
    return x**(1/2)


x = sy.Symbol("x")
print(sy.integrate(f(x)-g(x), (x, 0, 2)))

Output:

0.781048583502540

Example 2:

import matplotlib.pyplot as plt
import numpy as np


def f(x):
    return x**2


def g(x):
    return x**(1/2)


x = np.linspace(0, 2, 1000)
plt.plot(x, f(x))
plt.plot(x, g(x))
plt.fill_between(x, f(x), g(x), where=[(x > 0) and (x < 2) for x in x])
plt.show()

Output: