## Integrating functions in python

Posted February 02, 2013 at 09:00 AM | categories: python, math | tags:

Updated February 27, 2013 at 02:54 PM

**Problem statement**

find the integral of a function f(x) from a to b i.e.

$$\int_a^b f(x) dx$$

In python we use numerical quadrature to achieve this with the scipy.integrate.quad command.

as a specific example, lets integrate

$$y=x^2$$

from x=0 to x=1. You should be able to work out that the answer is 1/3.

from scipy.integrate import quad def integrand(x): return x**2 ans, err = quad(integrand, 0, 1) print ans

0.333333333333

## 1 double integrals

we use the scipy.integrate.dblquad command

Integrate \(f(x,y)=y sin(x)+x cos(y)\) over

\(\pi <= x <= 2\pi\)

\(0 <= y <= \pi\)

i.e.

\(\int_{x=\pi}^{2\pi}\int_{y=0}^{\pi}y sin(x)+x cos(y)dydx\)

The syntax in dblquad is a bit more complicated than in Matlab. We have to provide callable functions for the range of the y-variable. Here they are constants, so we create lambda functions that return the constants. Also, note that the order of arguments in the integrand is different than in Matlab.

from scipy.integrate import dblquad import numpy as np def integrand(y, x): 'y must be the first argument, and x the second.' return y * np.sin(x) + x * np.cos(y) ans, err = dblquad(integrand, np.pi, 2*np.pi, lambda x: 0, lambda x: np.pi) print ans

-9.86960440109

we use the tplquad command to integrate \(f(x,y,z)=y sin(x)+z cos(x)\) over the region

\(0 <= x <= \pi\)

\(0 <= y <= 1\)

\(-1 <= z <= 1\)

from scipy.integrate import tplquad import numpy as np def integrand(z, y, x): return y * np.sin(x) + z * np.cos(x) ans, err = tplquad(integrand, 0, np.pi, # x limits lambda x: 0, lambda x: 1, # y limits lambda x,y: -1, lambda x,y: 1) # z limits print ans

2.0

## 2 Summary

scipy.integrate offers the same basic functionality as Matlab does. The syntax differs significantly for these simple examples, but the use of functions for the limits enables freedom to integrate over non-constant limits.

Copyright (C) 2013 by John Kitchin. See the License for information about copying.