SciPy Integration (original) (raw)

Last Updated : 5 Jul, 2025

**Integration is a fundamental concept in calculus used to calculate areas under curves, volumes and in solving differential equations. In Python, the **SciPy library provides tools to perform both definite and indefinite integration using **scipy.integrate module.

Finding Integration using scipy.integrate

Numerical integration means calculating the value of an integral using approximation methods instead of solving it analytically. Common methods in scipy.integrate are:

• **quad : General purpose integration (for most 1D integrals)
• **dblquad : Double integration (for 2D integrals)
• **nquad : Multi-dimensional integration (n-fold integrals)
• **fixed_quad : Uses fixed order Gaussian quadrature
• **cumulative_trapezoid : Cumulative trapezoidal integration
• **simpson : Simpson’s rule (more accurate than trapezoidal rule)

Let's explore Examples of each method to understand it better.

**1. quad

**quad function computes definite integral of a function with respect to a single variable over given interval, points can be +infinite or - infinite to indicate infinite limits.

It returns two values:

• The integrated result
• An estimate of the absolute error

**Example:

Python `

from scipy.integrate import quad

def f(x): return 3 * x**2 + 1

I, err = quad(f, 0, 1) print(I) print(err)

`

**Output

2.0
2.220446049250313e-14

**2. dblquad

**dblquad function computes double integral of a function with two variables over a rectangular or curved region.

It takes:

• The integrand function (in terms of x and y)
• Limits for the outer variable (x)
• Lower and upper limits for the inner variable (y) as functions of x

**Example:

Python `

from scipy.integrate import dblquad

A = dblquad(lambda x, y: x * y, 0, 0.5,
lambda x: 0, lambda x: 1 - 2*x)
print(A)

`

**Output

(0.010416666666666668, 4.101620128472366e-16)

**3. nquad

**nquad function computes integrals of functions with multiple variables. It is a flexible method for performing n-dimensional integration over specified limits.

**Example:

Python `

from scipy.integrate import nquad

def f(x, y, z): return x * y * z

I = nquad(f, [[0, 1], [0, 5], [0, 5]]) print(I)

`

**Output

(78.12499999999999, 8.673617379884033e-13)

**4. fixed_quad

**fixed_quad function performs definite integration using Gaussian quadrature with a fixed number of points (defined by the n parameter).

**Example:

Python `

from scipy import integrate

def func(x): return 3 * x**3

B = integrate.fixed_quad(func, 1.0, 2.0, n=2) print(B)

`

**Output

(11.25), None

**5. cumulative_trapezoid

**cumulative_trapezoid() function computes cumulative integral of discrete data using the trapezoidal rule. Returns the running total of area showing how the integral accumulates step by step.

**Example:

Python `

from scipy.integrate import cumulative_trapezoid import numpy as np

x = np.arange(0, 5) y = np.arange(0, 5)

S = cumulative_trapezoid(y, x) print(S)

`

**Output

[0.5 2. 4.5 8. ]

**6. simpson

**simpson() function estimates area under a curve using Simpson’s rule. It works with discrete data points and gives more accurate results than the trapezoidal rule especially for smooth curves.

**Example:

Python `

import numpy as np from scipy.integrate import simpson

x = np.arange(0, 5) y = np.arange(0, 5)

S = simpson(y, x) print(S)

`