Integration of a function that is built by a loop

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What is the Integral of
for k = 1 :500
y = y +(sin( 2*k* x))./ k
end
in the bounderies of zero to pi ?

Respuesta aceptada

Bjorn Gustavsson
Bjorn Gustavsson el 6 de Mayo de 2020
You should do two things!
1, write that function definition into a function, then integrate it from 0 to pi. Use the integral or quadgk functions for the integration.
2, solve the more general problem: for n equal to 1, 2, 3, to 500.
HTH
  3 comentarios
Bjorn Gustavsson
Bjorn Gustavsson el 15 de Mayo de 2020
Herr Doktor Auf der Heyde,
Please be adviced to calculate the integral for an arbitrary term sin(2*n*x)/n from zero to π analytically by hand. Y is a sum of terms of simple continous functions on the intervall, therefore the integral of Y is equal to the sum of the integral of the terms.
I made a little illustrationing of this using this function:
function y = fivehundredsines( x,n_sines )
%UNTITLED2 Summary of this function goes here
% Detailed explanation goes here
if nargin < 2 || isempty(n_sines)
n_sines = 500;
end
y = 0;
for k = 1:n_sines
y = y +(sin( 2*k* x))./ k;
end
That one can run with:
x = linspace(0,pi,1001);
n = 1;
clf
Y =fivehundredsines(x,n);plot(x,Y),title([trapz(x,Y),integral(@(x)fivehundredsines(x,n),0,pi)]),xlabel(n),n = 2*n; hold on
Y =fivehundredsines(x,n);plot(x,Y),title([trapz(x,Y),integral(@(x)fivehundredsines(x,n),0,pi)]),xlabel(n),n = 2*n; hold on
Y =fivehundredsines(x,n);plot(x,Y),title([trapz(x,Y),integral(@(x)fivehundredsines(x,n),0,pi)]),xlabel(n),n = 2*n; hold on
Y =fivehundredsines(x,n);plot(x,Y),title([trapz(x,Y),integral(@(x)fivehundredsines(x,n),0,pi)]),xlabel(n),n = 2*n; hold on
HTH
Wolfgang Dr. Auf der Heyde
Wolfgang Dr. Auf der Heyde el 19 de Mayo de 2020
Pefect solution with minor modifications. The question arose from the Fejer_Jackson_Grönwall-Ungleichung-Math.Annalen1912-revisited www.mathe-kalender.de Prof.E.Wegert et.al.TU-Freiberg, you also may see www.complex-pictures.com Wolfg.AdH

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