# Definite integrals in Octave

In this lesson we see how to calculate the definite integral of a mathematical function in Octave with some practical examples.

**What do you need? **You must have already installed the Symbolic module on Gnu Octave.

Go to the Octave command line.

Define the variable symbol you want to use using the command **syms**. For example the variable x.

syms x

Now calculate the definite integral of the function f(x)=x^{2} in the interval (1,3)

$$ \int_1^3 x^2 \ dx $$

Type the command **int()** indicating the function x^{2} as the first parameter.

In the Octave syntax, the exponentiation is written **x**2**.

Write as the second parameter the lower bound of integration (1) and as the third parameter the upper bound of integration (3). Then hit enter.

int(x**2,1,3)

The result is the definite integral of x^{2} in the interval (1,3)

ans = (sym) 26/3

In this case the definite integral is equal to 26/3 that is about 8.6

**Verifica**. The integral of x^{2} in the range (1,3) is 26/3 because its primitive function is x^{3}/3 $$ \int_1^3 x^2 \ dx = [\frac{x^3}{3}]_1^3 = \frac{3^3}{3} - \frac{1^3}{3} = \frac{3^3-3^1}{3} = \frac{27-1}{3} = \frac{26}{3} $$

## An alternative method

There is also another way to calculate the definite integral of a function without using Symbolic.

Create the integral function in Octave as an **anonymous function** and calculate the integral defined by the function **quad()**

**quad(function name, a,b)**

For example, if the integral is

$$ \int_1^3 x^2 \ dx $$

The function of integral is f(x)=x^{2}

Write the anonymous function in Octave

>> f = @(x) x**2

To calculate the definite integral in the range from 1 to 3 use the function quad()

>> quad(f,1,3)

The quad function calculates the area under the function graph f(x)=x^{2} in the interval (1,3).

ans=8.6667

The definite integral is equal to 8.6667.

**Note**. It is the same result obtained in the previous example. $$ \int_1^3 x^2 \ dx = \frac{26}{3} = 8.667 $$