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# Integral and double Integral

## Integral

Matlab Integral is useful in finding areas under the curves. It is the reverse of differentiation in calculus and hence the functions are integrated by finding their anti-derivatives.

## Integrals are of 2 types:

1. Indefinite integrals (Integrals without limits)

2. Definite integrals (Integrals with limits)

### Syntax

Let us now understand the syntax of ‘integral function’ in MATLAB:

``A = integral (Fx, Xminimum, Xmaximum)``

### Explanation:

1. ‘integral function’ will calculate the numeric integration of input function ‘Fx’

2. ‘Xminimum’ and ‘Xmaximum’ will be used as a minimum and maximum limits for integration respectively

3. If we want to use more specific options for integral, we can use the syntax:

``A = integral (Fx, Xminimum, Xmaximum, Name, Value)``

## Examples to Implement Matlab Integral

Let us now understand how the code for ‘integral function’ looks like in MATLAB with the help of various examples:

## Example 1:

In this example, we will use a simple polynomial function of degree 2 and will integrate it between the limits 0 to 4. We will follow the following 2 steps:

Step 1: Create the function of degree 2 in MATLAB

Step 2: Use the integral function to calculate the integration

## Code:

``````syms x
[Initializing the variable ‘x’] Fx = @(x) 4*x.^2
[Creating the polynomial function of degree 2] A = integral (Fx, 0, 4)
[Passing input function ‘Fx’ and the required limits to the ‘integral function’] [Mathematically, the integral of 4*x ^ 2, between the limits 0 to 4 is 85.3333]``````

## Output:

Explanation: As we can see in the output, we have obtained integral of our input function ‘Fx’ as 85.3333 using ‘integral function’, which is the same as expected by us.

## Example 2:

In this example, we will use a polynomial function of degree 4 and will integrate it between the limits 0 to 2. We will follow the following 2 steps:

Step 1: Create the function of degree 4 in MATLAB

Step 2: Use the integral function to calculate the integration

## Code:

``````syms x
[Initializing the variable ‘x’] Fx = @ (x) (4 * x.^4 + x.^3 -2 * x.^2 +1)
[Creating the polynomial function of degree 4] A = integral (Fx, 0, 2)
[Passing input function ‘Fx’ and the required limits to the ‘integral function’] [Mathematically, the integral of 4 * x. ^ 4 + x. ^ 3 -2 * x. ^ 2 +1, between the limits 0 to 2 is 26.2667]``````

## Output:

Explanation: As we can see in the output, we have obtained integral of our input function ‘Fx’ as 26.2667 using ‘integral function’, which is the same as expected by us.

## Example 3:

In this example, we will learn how to integrate a function between the limits 0 and infinity. For this example, we will use a function which is a combination of logarithmic and exponential functions. The code will comprise of the following 2 steps:

Step 1: Create a function containing logarithmic and exponential functions

Step 2: Use the integral function to calculate the integration

## Code:

``````syms x
[Initializing the variable ‘x’] Fx = @(x) exp(-x. ^3). * log(2 * x). ^3;
[Creating the function containing the exponential and logarithmic functions] A = integral (Fx, 0, inf)
[Passing input function ‘Fx’ and the required limits to the ‘integral function’. Note that we have passed ‘inf’ which signifies infinity, as the upper limit] [Mathematically, the integral of exp (-x. ^3). * log (2 * x). ^3, between the limits 0 to infinity is         -2.9160]``````

## Output:

Explanation: As we can see in the output, we have obtained integral of our input function ‘Fx’ as -2.9160 using ‘integral function’, which is the same as expected by us.

## Example 4:

In this example, we will learn how to use the syntax A = integral (Fx, Xminimum, Xmaximum, Name, Value)

For this example, we will use a vector function which is of the form [log(x) log(2x) log (3x) log (4x)]. The code will comprise of the following 2 steps:

Step 1: Create a function containing vector values

Step 2: Use the integral function to calculate the integration and add a ‘name-value pair’ argument

## Code:

``````syms x
[Initializing the variable ‘x’] Fx = @(x) log((1 : 4) * x);
[Creating the function containing vector values] A = integral(Fx, 0, 2, 'ArrayValued', true)
[Passing input function ‘Fx’ and the required limits to the ‘integral function’. Note that we have passed ‘ArrayValued’, ‘true’, as the name value pair; which is used to calculate the integral of vector values]``````

Output:

Explanation: As we can see in the output, we have obtained integral of all the vector values in our array using integral function and ‘name-value pair’ argument.

# Double Integral:

## Double Integral

Matlab Double Integral is the extension of the definite integral. In double integral, the integration is performed for functions with 2 variables. In its simplest form, integration for functions with 1 variable is done over a 1-Dimensional space, likewise, integration of functions with 2 variables is done over a 2-D space. In MATLAB, we use ‘integral2 function’ to get the double integration of a function.

## Syntax:

Let us now understand the syntax of integral2 function in MATLAB:

``````I = integral2 (Func, minX, maxX, minY, maxY)
I = integral2 (Func, minX, maxX, minY, maxX, Name, Value)``````

## Explanation:

I = integral2 (Func, minX, maxX, minY, maxY) will integrate the function ‘Func’ (here ‘Func’ is a function of 2 variables X and Y) over the region minX ≤ X ≤ maxX and minY ≤ Y ≤ maxY

I = integral2 (Func, minX, maxX, minY, maxX, Name, Value) can be used to pass more options to the integral2 function. These options are passed as paired arguments

## Examples to Implement Matlab Double Integral

Let us now understand the code to calculate the double integral in MATLAB using ‘integral2 function’.

## Example 1:

In this example, we will take a function of cos with 2 variables ‘x’ and ‘y’. We will follow the following 2 steps:

Step 1: Create a function of x and y

Step 2: Pass the function and required limits to the integral2 function

## Code:

``````Func = @ (x,y) (x + cos (y) + 1)
[Creating the cos function in ‘x’ and ‘y’] I = integral2 (Func, 0, 1, 0, 2)
[Calling the integral2 function and passing the desired limits as 0 <= x <= 1; 0 <= y <= 2)] [Mathematically, the double integral of x + cos (y) + 1 is 3.9093]``````

## Output:

Explanation: As we can see in the output, we have obtained double integral of our input function as 3.9093, which is the same as expected by us.

## Example 2:

In this example, we will take a function of sin with 2 variables ‘x’ and ‘y’. We will follow the following 2 steps:

Step 1: Create a sin function of x and y

Step 2: Pass the function and required limits to the integral2 function

## Code:

``````Func = @(x,y) (sin(y) + x.^3 + 2)
[Creating the sin function in ‘x’ and ‘y’] I = integral2 (Func, 0, 1, 0, 2)
[Calling the integral function and passing the desired limits as 0 <= x <= 1; 0 <= y <= 2)] [Mathematically, the double integral of sin(y) + x.^3 + 2 is 5.9161]``````

## Output:

Explanation: As we can see in the output, we have obtained double integral of our input function as 5.9161, which is the same as expected by us.

## Example 3:

In this example, we will take a polynomial function of x and y. We will follow the following 2 steps:

Step 1: Create a polynomial function of x and y

Step 2: Pass the function and required limits to the integral2 function

## Code:

``````Func = @(x,y) (35*y.^3 - 15*x)
[Creating the polynomial function in ‘x’ and ‘y’] I = integral2 (Func, 0, 1, 0, 2)
[Calling the integral function and passing the desired limits as 0 <= x <= 1; 0 <= y <= 2)] [Mathematically, the double integral of 35*y.^3 - 15*x is 125]``````

## Output:

Explanation: As we can see in the output, we have obtained double integral of our input function as 125, which is the same as expected by us.

## Example 4:

In this example, we will take a polynomial function of x and y and of degree 3. We will follow the following 2 steps:

Step 1: Create a polynomial function of x and y and degree 3

Step 2: Pass the function and required limits to the integral2 function

## Code:

``````Func = @(x,y) ((20*y.^3) + 3*x.^2)
[Creating the polynomial function in ‘x’ and ‘y’] I = integral2 (Func, 0, 1, 0, 2)
[Calling the integral function and passing the desired limits as 0 <= x <= 1; 0 <= y <= 2)] [Mathematically, the double integral of (20*y.^3) + 3*x.^2 is 82]``````

## Output:

Explanation: As we can see in the output, we have obtained double integral of our input function as 82, which is the same as expected by us.

## Example 5:

In this example, we will take a polynomial function with the division. We will follow the following 2 steps:

Step 1: Create a polynomial function of x and y and with division

Step 2: Pass the function and required limits to the integral2 function

## Code:

``````Func = @(x,y) 1./(sqrt(x.^3 + y.^2))
[Creating the polynomial function in ‘x’ and ‘y’ and division] I = integral2 (Func, 0, 1, 0, 2)
[Calling the integral function and passing the desired limits as 0 <= x <= 1; 0 <= y <= 2)] [Mathematically, the double integral of 1./(sqrt(x.^3 + y.^2))  is 2.9012]``````

## Output:

Explanation: As we can see in the output, we have obtained double integral of our input function as 2.9012, which is the same as expected by us.