Department of Computing
MATH 333: Numerical Analysis
Class: BSCS-9ABC
Lab 13: Numerical Integration
Date: May 2nd, 2023
Time: 10:00 - 1:00
Lab Engineer: Anum Asif
Name: Hassan Haroon
CMS: 339817
Lab 13: Numerical Integration
Introduction
Matlab represents polynomials with a vector of coefficients. The length of the vector will always
be one more than the order of the polynomial
Objectives
The purpose of this lab is to get familiar with Numerical Integration.
Tools/Software Requirement
Matlab
Description
Need of Numerical Integration: For evaluating the definite integral of a function that has no explicit
anti-derivatives or whose anti-derivatives is not easy to obtain.
Lab Task:
1.
Output:
Code:
% Data Points
x_values = 0:2:20;
y_values = [0 16 29 40 46 51 32 18 8 3 0];
% Interval
interval = x_values(2) - x_values(1);
% Get the number of components
num_components = length(x_values);
simpson_1_3rd = 0;
% Loop through each set and apply Simpson's 1/3rd rule
for i = 1:2:num_components-2
simpson_1_3rd = simpson_1_3rd + (interval/3) * (y_values(i+2) + 4*y_values(i+1) +
y_values(i));
end
disp(['The result obtained using Simpson''s 1/3rd rule is: ',
num2str(simpson_1_3rd)])
2.
Suppose that the upward force of air resistance on a falling object is proportional to the
square of the velocity. For this case velocity can be computed as
𝑣(𝑡) = √
𝑔𝑚
𝑔𝑐𝑑
tanh (√
𝑡)
𝑐𝑑
𝑚
Where 𝑐𝑑 = a second-order drag coefficient. If = 9.8 𝑚/𝑠 2 , 𝑚 = 68.1 𝑘𝑔 and 𝑐𝑑 = 0.25𝑘𝑔/𝑚,
use trapezoidal rule to determine how far the object falls in 10𝑠. Use a sufficiently high 𝑛 that
you get eight significant digits of accuracy.
Output:
Code:
% Initialize gravity, mass, and drag coefficient
gravity = 9.8;
mass = 68.1;
drag_coefficient = 0.25;
% Start time
time_start = 0;
% End time
time_end = 10;
% Number of intervals
num_intervals = 1000000;
% Calculate the interval size
interval_size = (time_end - time_start) / num_intervals;
% Function for velocity
velocity = @(t) sqrt((gravity * mass / drag_coefficient)) * tanh(sqrt((gravity *
drag_coefficient) / mass) * t);
% Initialize distance
distance = 0;
% Apply Trapezoidal rule
for i = 1:num_intervals
t1 = time_start + (i-1) * interval_size;
t2 = time_start + i * interval_size;
distance = distance + interval_size / 2 * (velocity(t1) + velocity(t2));
end
% Display the result
disp(['The distance fallen in 10 seconds is: ', num2str(distance), ' meters']);
Deliverables
Submit single word file with matlab code and screen shot of Output.