Tutorial: Integration

Tutorial: Integration

Tutorial: Integration Solver- student

Exercise - MATLAB

Estimate the velocity from the dataset of acceleration

clear all
x=[0 5 10 15 20 25 30 35 40 47 50 54 60];
y=[0 3 8 20 33 42 40 48 60 12 8 4 3];
N=length(x);
plot(x,y,'.-');

% Matlab function
I_matlab = trapz(x,y);    

Download the tutorial source file and fill in the blanks.

Run the code and validate your answer

• MATLAB tutorial source file : TU_Integration_Matlab.zip

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Exercise - C Programming

Go to \tutorial Directory

• e.g ) C:\Users\yourID\source\repos\NP\tutorial

Create a new empty project in Visual Studio Community. Name the project as TU_Integration

• e.g ) C:\Users\yourID\source\repos\NP\tutorial\TU_Integration

Create a new C/C++ source file for main()

• Name the source file as TU_integration_main.cpp

• Paste from the following code: TU_integration_student

Download the answer report files: Answer Report

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Part 1 : Integration of Discrete Points

Create a function for numerical differentiation from a set of data. Read the instructions in the source code.

Exercise 1: Trapezoid (15 minutes)

Create Trapezoidal method for discrete data inputs. Upload the result in LMS.

$$I(f) = \frac{1}{2}\sum\limits_{i = 0}^{N - 1} {\left[ {f\left( {{x_i}} \right) + f\left( {{x_{i + 1}}} \right)} \right]({x_{i + 1}} - {x_i})}$$

double trapz (double x[ ], double y[ ], int m);

• In the report, screen capture the output window and paste your code

• Use 1D array type with dataset length m.

• intervals= N, # dataset=N+1=m, The ranges are x[0] to x[m-1]

Declare and define your functions in your header files, located in \include folder

• function definitions: myNP.h

• function declaration: myNP.cpp

Exercise 2: Simpson13 (15 minutes)

Create Simpson13 method for discrete data inputs. Upload the result in LMS.

• In the report, screen capture the output window and paste your code

• Use Simpson 13 method :

• N even numbers, same intervals, from a(=x0) to b(=xN), h=(b-a)/N.

• The interval should be h=(b-a)/N

• Defined in myNM.cpp source file

double simpson13(double x[ ], double y[ ], int m);

Simpson 13 method :

$$I = \frac{h}{3}[f({x_0}) + 4\sum\limits_{i = 1,3,5}^{N - 1} {f\left( {{x_i}} \right) + 2\sum\limits_{k = 2,4,6}^{N - 2} {f\left( {{x_k}} \right)} + } f({x_N})]$$

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Part 2: Integration from a Function

Exercise 3 : (20min)

Create Simpson13() when a function is given as the input. Upload the result in LMS

double myFunc (const double x);  // in main.cpp
double integral(double func(const double x), double a, double b, int n);  // in myNM.h

• In the report, screen capture the output window and paste your code

• Use Simpson 13 method :

• N even numbers, same intervals, from a(=x0) to b(=xN), h=(b-a)/N.

• The interval should be h=(b-a)/N

• Defined in myNM.cpp source file

Create the following function that calls myFunc() . The true answer is PI/2

$$I(f) = \int_{ - 1}^1 {\sqrt {1 - {x^2}} dx}$$