Trying to integrate sinx using the integrate function but when I plot the integral, instead of plotting from -1 to 1 it plots from 0 to 2. Does anyone know why this happens?

I am using the "integrate" function to plot the integral of sin(x) [which is -cos(x)]. When I plot the integral it should plot from -1 to 1 but instead it is shifting the integral from 0 to 2. In the code below I show what I did. The @myFunInt is the following

function fval = myFunInt(x)
fval = sin(x);
end

and the main code is

clear
a=0;            % lower limit
b=30;           % upper limit
n=1000;         % subintervals
h = (b-a)/n;    % Spacing

x = 0:30;
int = zeros(1,n+1);

for j = 0:n    
    x_j=a+j*h;      % x values are being allocated in the empty array
    x(:,j+1)=x_j;
    fun = @myFunInt;
    y=integral(fun,a,x(:,j+1));
    int(:,j+1)=y;
end
plot(x,int);

Note that this code works with any other function exept than integrading cos(x) and -sin(x).

Thank you for your input.

Answer 1

[John D'Errico]

Actually, the integral of sin(x) is NOT -cos(x). This is a mistake that people frequently make. The integral is -cos(x), PLUS a constant of integration. That is, the indefiinite integral:

int(sin(x),x) == -cos(x) + C

Where C can be any constant number. Why is this important? Suppose you think of this as a differential equation:

df/dx = sin(x)

Now, you want to integrate that ordinary differential equation, over the interval [a,b] = [5,30]. Again, we will find the general solution as

f(x) = -cos(x) + C

However, then when you solve an ODE, you consider the initial conditions to resolve the constant. What is f(a)? In your code, you implicitly defined f(a) to be ZERO. So, implicitly, you resolved the constant C as:

C = cos(5)
C =
        0.283662185463226

Now, given ths plot you created, I'll overlay the function -cos(x)+C.

plot(x,int);
hold on
xint = linspace(5,30,100);
plot(xint,-cos(xint) + C,'ro')

As you can see, the curve you generated (in blue) and the red dots I plotted overlay perfectly.

You need to remember that constant of integration, or it will hurt you when you are not looking. As far as your code having worked for you on other problems, well you may have just gotten lucky. Everything would look perfectly ok as a result of your code, IF it turned out that the initial condition of f(a)=0 just happened to have been correct. And then you ran into this problem, where that initial condition turned out to have been not what you expected.

[Guillermo Naranjo]

Thank you very much John.

I was not taking into account the indefinite integral. I was just assuming since I had a limited range it would behave in a different manner. Thank you for your reply.

Best,

Guillermo

Answer 2

[Geoff Hayes]

Guillermo - I'm guessing it is the y-axis limits that you want to be between -1 and 1? Why is the minimum value, a, fixed but the maximum is variable? Why the interval [5,30]? Is the units for this interval supposed to be radians, degrees, or does it not matter?

Note that if I use a and b as

a=0;            % lower limit
b=2*pi;         % upper limit

Then the figure is drawn as

which has the interval [0,2] which is what you describe. Is this expected? I think so because when you integrate like

y=integral(fun,a,x(:,j+1));

then y is the area under the curve...so on each iteration of your loop, you are calculating the area from 0 to something a little larger (a function of j and h). When x is pi, then the area will be two (since area from 0 to pi/2 is one).

[Guillermo Naranjo]

Geoff,

Thank you for your fast reply. The limits are arbitrary for what I am trying to accomplish. In this case, I am plotting it in radians which is why i am not using sind(x).

I see what you did. However, if you try integrating cos(x) instead of sin(x), the limits in the y-axis will be from -1 to 1 as expected.

What i am trying to show is that no matter what limits I use, if I am integrating a cos(x) or sin(x) with amplitudes of -1 to 1. Then the integrated function should inherently have the same limits.

What I dont understand is why it shifts when you try to integrate sin(x) or -cos(x).

I wrote a script that solves the integral using the Simpson's rule and I get the same effect as when I use the "integrate" function.

Also, -cos(pi/2) is zero so the value in the plot you show technically does not represent the integral of sin(x). It seems that what the plot shows is y = -cos(x)+1 from [0 2pi].

Thanks again for your input.

[Geoff Hayes]

Also, -cos(pi/2) is zero so the value in the plot you show technically does not represent the integral of sin(x).

Yes Guillermo, but on each iteration of your loop, you are calculating

y=integral(fun,a,x(:,j+1));

So isn't y the area under the curve from a to x(:,j+1)? And so on each iteration of your loop, you are calculating an area which increases from 0 to pi (using my interval) and then decreases from pi to 0. You are then plotting the area calculated on each iteration...you are not plotting the integral of sin(x).

See https://www.mathwords.com/a/area_under_a_curve.htm and https://girlsangle.wordpress.com/2013/07/31/why-does-the-area-under-one-hump-of-a-sine-curve-exactly-equal-2/

[Geoff Hayes]

However, if you try integrating cos(x) instead of sin(x), the limits in the y-axis will be from -1 to 1 as expected.

Correct but again you have calculated and are plotting areas under the curve, and for cos, these limits make sense given its curve.

[Guillermo Naranjo]

Geoff,

Thank you again for your answer. You are correct, when I integrate from 0 to pi and then from pi to zero the total area under the curve is 0. So yes it depends on the interval I am calculating the area under the curve.

I thought of this which is why I included the term

int(:,j+1)=y;

What this does is for every sub interval

a=0;            % lower limit
b=pi;           % upper limit
n=1000;         % subintervals
h = (b-a)/n;    % Spacing

I will save the current area and plot that value which I understand would give me the integral of sin(x). Is this not right?

[Guillermo Naranjo]

Geoff,

After reading Jonh's reply I realized I did not take into account the constant the integral of sin(x). This is what is causing my integral to shift.

Thanks again for everything.