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    MATLAB Examples - Numerical Integration

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    Danh Bich Do
    The document discusses numerical integration in MATLAB. It introduces the concept of approximating the area under a curve by dividing it into rectangles and summing their areas. As an example, it calculates the integral of y=x^2 from 0 to 1 using the trapezoid rule in MATLAB. It then demonstrates using built-in functions like quad() and quadl() for numerical integration. Finally, it shows how to calculate the integral of a polynomial function from -1 to 1 using the polyint() function.

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    MATLAB Examples - Numerical Integration

    Uploaded by

    Danh Bich Do
    64 views17 pages
    The document discusses numerical integration in MATLAB. It introduces the concept of approximating the area under a curve by dividing it into rectangles and summing their areas. As an example, it calculates the integral of y=x^2 from 0 to 1 using the trapezoid rule in MATLAB. It then demonstrates using built-in functions like quad() and quadl() for numerical integration. Finally, it shows how to calculate the integral of a polynomial function from -1 to 1 using the polyint() function.

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    The document discusses numerical integration in MATLAB. It introduces the concept of approximating the area under a curve by dividing it into rectangles and summing their areas. As an example, it calculates the integral of y=x^2 from 0 to 1 using the trapezoid rule in MATLAB. It then demonstrates using built-in functions like quad() and quadl() for numerical integration. Finally, it shows how to calculate the integral of a polynomial function from -1 to 1 using the polyint() function.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    64 views17 pages

    MATLAB Examples - Numerical Integration

    Uploaded by

    Danh Bich Do
    The document discusses numerical integration in MATLAB. It introduces the concept of approximating the area under a curve by dividing it into rectangles and summing their areas. As an example, it calculates the integral of y=x^2 from 0 to 1 using the trapezoid rule in MATLAB. It then demonstrates using built-in functions like quad() and quadl() for numerical integration. Finally, it shows how to calculate the integral of a polynomial function from -1 to 1 using the polyint() function.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
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    of 17

    MATLAB

    Examples
    Numerical Integration

    Hans-Petter Halvorsen
    Integration

    The integral of a function 𝑓(𝑥) is denoted as:

    '
    % 𝑓 𝑥 𝑑𝑥
    (
    Integration
    Given the function:
    𝑦 = 𝑥+

    We know that the exact solution is:


    ( -
    𝑎
    % 𝑥 + 𝑑𝑥 =
    / 3
    The integral from 0 to 1 is:
    1
    1
    +
    % 𝑥 𝑑𝑥 = ≈ 0.3333
    / 3
    Numerical Integration

    An integral can be seen as the area under a curve.


    Given 𝑦 = 𝑓(𝑥) the approximation of the Area (𝐴) under the curve can be found dividing the area up into
    rectangles and then summing the contribution from all the rectangles (trapezoid rule):
    Example: Numerical Integration
    We know that the exact solution is:

    We use MATLAB (trapezoid rule):


    x=0:0.1:1;
    y=x.^2;
    plot(x,y)

    % Calculate the Integral:


    avg_y=y(1:length(x)-1)+diff(y)/2;
    A=sum(diff(x).*avg_y)

    A = 0.3350
    Students: Try this example
    Example: Numerical Integration
    We know that the exact solution is:

    In MATLAB we have several built-in functions we can use for numerical integration:
    clear
    clc
    close all

    x=0:0.1:1;
    y=x.^2;

    plot(x,y)

    % Calculate the Integral (Trapezoid method):


    avg_y = y(1:length(x)-1) + diff(y)/2;
    A = sum(diff(x).*avg_y)

    % Calculate the Integral (Simpson method):


    A = quad('x.^2', 0,1)

    % Calculate the Integral (Lobatto method):


    A = quadl('x.^2', 0,1)
    Numerical Integration
    Given the following equation:

    𝑦 = 𝑥 - + 2𝑥 + − 𝑥 + 3

    • We will find the integral of y with respect to x, evaluated from -1


    to 1
    • We will use the built-in MATLAB functions diff(), quad() and
    quadl()
    Numerical Integration – Exact Solution
    The exact solution is:
    '
    ' : - +
    - +
    𝑥 2𝑥 𝑥
    𝐼 = % (𝑥 + 2𝑥 − 𝑥 + 3)𝑑𝑥 = + − + 3𝑥 <
    ( 4 3 2
    (
    1 : 2 - 1 +
    = 𝑏 − 𝑎 + 𝑏 − 𝑎 − 𝑏 − 𝑎+ + 3(𝑏 − 𝑎)
    : -
    4 3 2

    𝑎 = −1 and 𝑏 = 1 gives:
    1 2 1 22
    𝐼 = 1−1 + 1+1 − 1−1 +3 1+1 =
    4 3 2 3
    Symbolic Math Toolbox
    We start by finding the Integral using the Symbolic Math Toolbox:
    clear, clc

    syms f(x)
    syms x

    f(x) = x^3 + 2*x^2 -x +3

    I = int(f)

    This gives: I(x) = x^4/4 + (2*x^3)/3 - x^2/2 + 3*x

    http://se.mathworks.com/help/symbolic/getting-started-with-symbolic-math-toolbox.html
    Symbolic Math Toolbox
    The Integral from a to b:
    clear, clc
    syms f(x)
    syms x
    f(x) = x^3 + 2*x^2 -x +3

    a = -1;
    b = 1;
    Iab = int(f, a, b)

    This gives: Iab = 22/3 ≈ 7.33

    http://se.mathworks.com/help/symbolic/getting-started-with-symbolic-math-toolbox.html
    clear, clc

    x = -1:0.1:1;

    y = myfunc(x);

    plot(x,y)

    % Exact Solution
    a = -1;
    b = 1;
    Iab = 1/4*(b^4-a^4 )+2/3*(b^3-a^3 )-
    1/2*(b^2-a^2 )+3*(b-a)

    % Method 1
    avg_y = y(1:length(x)-1) + diff(y)/2;
    A1 = sum(diff(x).*avg_y)
    MATLAB gives the following results:
    % Method 2
    A2 = quad(@myfunc, -1,1) Iab = 7.3333
    A1 = 7.3400
    % Method 3 A2 = 7.3333
    A3 = quadl(@myfunc, -1,1)
    A3 = 7.3333
    Integration on Polynomials
    Given the following equation:

    𝑦 = 𝑥 - + 2𝑥 + − 𝑥 + 3

    Which is also a polynomial. A polynomial can be written on the


    following general form: 𝑦 𝑥 = 𝑎/ 𝑥 @ + 𝑎1 𝑥 @A1 + ⋯ + 𝑎@A1 𝑥 + 𝑎@

    • We will find the integral of y with respect to x, evaluated from -1 to 1


    • We will use the polyint() function in MATLAB
    clear The solution is:
    ans =
    clc 0.2500 0.6667 -0.5000 3.0000 0

    p = [1 2 -1 3]; The solution is a new polynomial:


    [0.25, 0.67, -0.5, 3, 0]

    polyint(p) Which can be written like this:

    0.25𝑥 : + 0.67𝑥 - − 0.5𝑥 + + 3𝑥

    We know from a example that the exact solution is:

    '
    ' : - +
    - +
    𝑥 2𝑥 𝑥
    % (𝑥 + 2𝑥 − 𝑥 + 3)𝑑𝑥 = + − + 3𝑥 <
    ( 4 3 2
    (

    → So wee se the answer is correct (as expected).


    Hans-Petter Halvorsen, M.Sc.

    University College of Southeast Norway


    www.usn.no

    E-mail: hans.p.halvorsen@hit.no
    Blog: http://home.hit.no/~hansha/

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