Integration of a function that is built by a loop

What is the Integral of

for k = 1 :500

y = y +(sin( 2*k* x))./ k

end

in the bounderies of zero to pi ?

Answer 1

[Bjorn Gustavsson]

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

[Wolfgang Dr. Auf der Heyde]

The problem to integrate for k=1:500 ; y=y + ((sin 2*k*x) / k ; end in the limits of 0 to pi ( or 0 to 2pi) is still not solved with the answer (1).

1. To write a function (routine) is not essentiell for the problem

Wollfgang AdH

[Bjorn Gustavsson]

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]

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