Example Problem: finding the mass of a wire with a scalar line integral

Example Problem: finding the mass of a wire with a scalar line integral.
I set up the integral, evaluate it in MATLAB, and also take a look at it graphically.

Here is the problem statement:

Let W represent a thin wire bend in the shape of a circle of radius 3 centered at the origin in the xy-plane. The density of the wire is given by ρ(x,y)= x^2+1 grams/cm at every point (x,y) along the wire. Find the mass of the wire.

So we can start with our equation for the scalar line integral
We insert our density as the function where m is total mass we're looking for and we know that ds becomes |r ⃗^' (t)| dt.

But we're not sure exactly what to plug in for a or b or what |r ⃗^' (t)| is yet.
So to find those we graph it out.
We need to find parametric equations for the x and y values as a function of time .
This takes a little creativity but as you may recall unit circle can be defined by parametric functions of sine and cosine for our x and y
So if we defined
x=cos(t)
and
y=sin(t)
this would give us a unit circle starting at the point
(1,0)
at
t=0=a
and going all the way around the circle through
t=2π=b
so those t values will be our a and b.

Our circle is scaled up from that by 3, so we can just multiply each component by 3, yielding
x=3cos(t)
and
y=3sin(t)
so r would be
r ⃗(t)= 〈3cos(t), 3sin(t)〉

But we need
|r ⃗^' (t)|

This isn't too hard to figure out but let's let MATLAB take it from here. But first, let me note that there are infinite combinations of sin and cos parameterizations that would work for this. The parameterizations could start at different places and go different speeds, but as we are parameterized to take one loop around the circle, we’ll get the same correct answer.

But let’s go with what we’ve got here and plug it into MATLAB

And start of by defining all the symbolic variables and let’s just define everything
syms x y r t rho

Then we can define the x we found
x=3*cos(t)

And the y we found
y=3*sin(t)

And combine those together for r
r=[x,y]

and we can also define rho as
rho=x^2+1

And finally plug those into our integral of our rho function x^2+1 times the magnitude of the derivative of r (and this is a common place to miss something) by using the norm and diff commands, wrt t, wrt t from t = 0 to 2pi
m=int(rho*norm(diff(r,t)),t,[0,2*pi])

This yields a tidy value of
33π
which is our final answer but let's do one last thing
I may be going too far with this but personally I need to understand things graphically so let's imagine we have our circle here in 3 space.
Now let's imagine our z axis corresponds to the density of our function, so at each of these point along our red circle we have a height over it in blue corresponding to how dense the wire is at that point.
Then that line integral we found, which was the total mass of the wire, corresponds graphically here to the area underneath this blue line,
So our line integral is still the area under a curve, It's just that in this case the height of that curve has the physical meaning of density.

So I hope this idea of a line integral is making sense to you, and we'll move on from here to take a look at line integrals over vector fields next.