With Chebfun it is easy to compute with parametrized curves in the plane. For example, the following lines define a curve $(x,y)$ as a pair of chebfuns in the variable $t$:

t = chebfun('t',[0,2*pi]);
b = 1; m = 7; a = (m-1)*b;
x = (a+b)*cos(t) - b*cos((a+b)/b*t);
y = (a+b)*sin(t) - b*sin((a+b)/b*t);

Such curves are called epicycloids, named by the Danish astronomer Ole Romer in 1674. Epicycloids can be produced by tracing a point on a circle which rolls out on a larger circle. Romer discovered that cog-wheels with epicycloidal teeth turned with minimum friction. This is what our epicycloid looks like:

fill(x,y,[.6 .6 1])
axis equal

Note that although this curve is not smooth, the functions $x(t)$ and $y(t)$ that parameterize it are smooth, so Chebfun has no difficulty representing them by global polynomials:

x
y
x =
chebfun column (1 smooth piece)
interval       length   endpoint values
[       0,     6.3]       53         6        6
Epslevel = 1.856760e-15.  Vscale = 7.386013e+00.
y =
chebfun column (1 smooth piece)
interval       length   endpoint values
[       0,     6.3]       52   8.9e-16 -8.9e-16
Epslevel = 1.716014e-15.  Vscale = 8.

With the following formula we can compute the area enclosed by the curve $(x,y)$:

format long
A = sum(x.*diff(y))
A =
1.759291886010285e+02

Let's compare this result with the exact area of the epicycloid, given (for integer $m$) by the formula

exact = pi*b^2*(m^2+m)
exact =
1.759291886010284e+02

Here is a more complicated curve (now defined as a single complex-valued chebfun rather than a pair of real-valued chebfuns):

z = exp(1i*t) + (1+1i)*sin(6*t).^2;
fill(real(z),imag(z),[.6 1 .6])
axis equal

Because this curve is a perturbed unit circle, with every perturbation occurring twice with opposite signs, the enclosed area should be equal to $\pi$, as is confirmed by Chebfun:

A = sum(real(z).*diff(imag(z)));
[ A ; pi ]
ans =
3.141592653589794
3.141592653589793

We can compute and plot the centroid (or center of mass) of this region as follows:

c = sum(diff(z).*z.*conj(z))/(2i*A);
hold on
plot(c,'r+','markersize',20)

If you use scissors to produce a piece of paper in this shape, it should remain balanced when placed on a vertical needle centered at the red cross. (If it doesn't, it's likely your handicraft precision isn't as good as Chebfun's!)