Matlab integration function

Integration is defined as the process of finding the anti derivative of a function. It is used to calculate area, volume, displacement, and many more. In this article, we will see how to perform integration on expressions in MATLAB.

There are two types of Integration:

• Indefinite integral: Let f(x) be a function. Then the family of all antiderivatives is called the indefinite integral of a function f(x) and it is denoted by ∫f(x)dx. The symbol ∫f(x)dx is read as the indefinite integral of f(x) with respect to x. Thus ∫f(x)dx= ∅(x) + C. Thus, the process of finding the indefinite integral of a function is called the integration of the function.
• Definite integrals: 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. It is denoted by ∫f(x)dx under the limit of a and b, 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.

Before moving to Integration, we first need to assign an expression to a variable in MATLAB which can be done by using the inline() function. It creates a function to contain the expression.

Syntax:

f = inline(expr, var)

Here f is the inline function created to contain the expression, expr can be any expression and var is the variable in that expression.

Now, after assigning the expression using the inline() function, we need to integrate the expression. This task can be performed using the int() function. The int() function is used to integrate expressions in MATLAB.

Syntax:

int(f,v)

Parameters:

Indefinite Integral

Indefinite integrals are those integrals that do not have any limit and containing an arbitrary constant.

Step-wise Approach:

Step 1: Use Inline function for the creation of the function for integration.

Step 2: Create a symbolic function.

Step 3: Use int to find out the integration.

Complete Code:

Output: 

Definite Integral

Definite integrals are those integrals that have an upper limit and a lower limit. Let’s take the above example and add the limits.

Step-wise Approach:

Step 1: Use the Inline function for the creation of the function for integration.

Step 2: Create a symbolic function.

Step 3: Use int to find out the integration and pass down the values of the lower limit, upper limit.

 Complete Code:

Output:

An integral is a mathematical measure that combines infinitesimal data points. Integrals have a broad range of applications in all engineering disciplines.

Types of Integrals

In general, integrals can be either definite or indefinite. Definite integrals represent functions with bounded upper and lower limits, whereas indefinite Integrals represent functions without limits.

The following example shows an indefinite integral:

$$I= \int3x^2 dx=x^3+c$$

where ‘c’ is a constant.

A definite integral for the same equation must have defined limits. For example, we can integrate the above equation with limits [-2, 2] as follows:

$$I= \int_{-2}^2 3x^2 dx=(2^3+c)-(-2^3+c)=16$$

You can use MATLAB^® and Symbolic Math Toolbox™ to calculate integrals numerically and symbolically.

Examples of Integral Applications

1. Area under curves:

You can calculate the area under two curves using integrals. For example, we define two curves,

$$x1=y^2-1$$

$$x2=1- y^2$$

and calculate the area under the curves as follows:

$$A=∫(x2-x1) dy=\frac{(-2y(y^2-3))}{3}$$

The area under the curve\('A’\) here is a function of \('y’\) because we did not specify limits. If we define the limits as \([-1, 1]\), the integral returns a value of:

$$A=8/3.$$

1. Volume of objects:

You can use integrals to calculate the volume of objects. For example, you can derive the volume of a sphere by starting with a function:

$$f(x)=√(r^2-x^2 )$$

which depicts a semi-circle with radius ‘r.’ Rotating this semi-circle around the x-axis would yield a sphere.

The area of the semi-circle would be

$$A=πf(x)^2$$

Integrating this area with limits [-r, +r] would give us the volume of the sphere:

$$V= ∫_{-r}^{+r} A dx=\frac{(4πr^3)}{3}$$

1. Velocity of a moving object:

You can find the velocity of an object by finding the definite integral of the object’s acceleration with respect to time, because acceleration is simply defined as the rate of change of velocity over time.

$$∆Vel= ∫Acc \; dt$$

Techniques to Calculate Integrals

You can calculate integrals numerically using techniques such as:

1. Simpson quadrature
2. Lobatto quadrature
3. Gauss-Kronrod quadrature

For more information on the numeric and symbolic calculations of integrals, see MATLAB^® and Symbolic Math Toolbox™.

Integration

If is a symbolic expression, then

attempts to find another symbolic expression, , so that   . That is, returns the indefinite integral or antiderivative of (provided one exists in closed form). Similar to differentiation,

uses the symbolic object as the variable of integration, rather than the variable determined by . See how works by looking at this table.

In contrast to differentiation, symbolic integration is a more complicated task. A number of difficulties can arise in computing the integral:

Nevertheless, in many cases, MATLAB can perform symbolic integration successfully. For example, create the symbolic variables

The following table illustrates integration of expressions containing those variables.

In the last example, , there is no formula for the integral involving standard calculus expressions, such as trigonometric and exponential functions. In this case, MATLAB returns an answer in terms of the error function .

If MATLAB is unable to find an answer to the integral of a function , it just returns .

Definite integration is also possible.

Here are some additional examples.

For the Bessel function () example, it is possible to compute a numerical approximation to the value of the integral, using the function. The commands

return

and the command

returns

Integration with Real Parameters

One of the subtleties involved in symbolic integration is the “value” of various parameters. For example, if a is any positive real number, the expression

is the positive, bell shaped curve that tends to 0 as x tends to ±∞. You can create an example of this curve, for a = 1/2.

However, if you try to calculate the integral

without assigning a value to a, MATLAB assumes that a represents a complex number, and therefore returns a piecewise answer that depends on the argument of a. If you are only interested in the case when a is a positive real number, use to set an assumption on :

Now you can calculate the preceding integral using the commands

This returns

Integration with Complex Parameters

To calculate the integral

for complex values of , enter

Use to clear the all assumptions on variables. For more information about symbolic variables and assumptions on them, see Delete Symbolic Objects and Their Assumptions.

The preceding commands produce the complex output

The function is defined as:

To evaluate at , enter

High-Precision Numerical Integration Using Variable-Precision Arithmetic

High-precision numerical integration is implemented in the function of the Symbolic Math Toolbox™. uses variable-precision arithmetic in contrast to the MATLAB function, which uses double-precision arithmetic.

Integrate by using both and . The function returns and issues a warning while returns the correct result.

For more information, see .

Main Content

Integration deals with two essentially different types of problems.

• In the first type, derivative of a function is given and we want to find the function. Therefore, we basically reverse the process of differentiation. This reverse process is known as anti-differentiation, or finding the primitive function, or finding an indefinite integral.

• The second type of problems involve adding up a very large number of very small quantities and then taking a limit as the size of the quantities approaches zero, while the number of terms tend to infinity. This process leads to the definition of the definite integral.

Definite integrals are used for finding area, volume, center of gravity, moment of inertia, work done by a force, and in numerous other applications.

Finding Indefinite Integral Using MATLAB

By definition, if the derivative of a function f(x) is f'(x), then we say that an indefinite integral of f'(x) with respect to x is f(x). For example, since the derivative (with respect to x) of x² is 2x, we can say that an indefinite integral of 2x is x².

In symbols −

f'(x²) = 2x, therefore,

∫ 2xdx = x².

Indefinite integral is not unique, because derivative of x² + c, for any value of a constant c, will also be 2x.

This is expressed in symbols as −

∫ 2xdx = x² + c.

Where, c is called an 'arbitrary constant'.

MATLAB provides an int command for calculating integral of an expression. To derive an expression for the indefinite integral of a function, we write −

For example, from our previous example −

MATLAB executes the above statement and returns the following result −

Example 1

In this example, let us find the integral of some commonly used expressions. Create a script file and type the following code in it −

When you run the file, it displays the following result −

Example 2

Create a script file and type the following code in it −

Note that the pretty function returns an expression in a more readable format.

When you run the file, it displays the following result −

Finding Definite Integral Using MATLAB

By definition, definite integral is basically the limit of a sum. We use definite integrals to find areas such as the area between a curve and the x-axis and the area between two curves. Definite integrals can also be used in other situations, where the quantity required can be expressed as the limit of a sum.

The int function can be used for definite integration by passing the limits over which you want to calculate the integral.

To calculate

we write,

For example, to calculate the value of we write −

MATLAB executes the above statement and returns the following result −

Following is Octave equivalent of the above calculation −

Octave executes the code and returns the following result −

An alternative solution can be given using quad() function provided by Octave as follows −

Octave executes the code and returns the following result −

Example 1

Let us calculate the area enclosed between the x-axis, and the curve y = x³−2x+5 and the ordinates x = 1 and x = 2.

The required area is given by −

Create a script file and type the following code −

When you run the file, it displays the following result −

Following is Octave equivalent of the above calculation −

Octave executes the code and returns the following result −

An alternative solution can be given using quad() function provided by Octave as follows −

Octave executes the code and returns the following result −

Example 2

Find the area under the curve: f(x) = x² cos(x) for −4 ≤ x ≤ 9.

Create a script file and write the following code −

When you run the file, MATLAB plots the graph −

The output is given below −

Following is Octave equivalent of the above calculation −

Main Content