Suppose I have a closed curve described by a parametrized function, for example $f(t) =[cos(\frac{1}{2}t)+\frac{1}{2}cos(t);sin(\frac{1}{2}t)+\frac{1}{2}sin(t)]$ with $(0< t <4\pi)$.
How can I find the center of mass of such a closed curve? 3blue1brown (https://www.youtube.com/watch?v=spUNpyF58BY&t=462s) plays a lot with this concept of a center of mass in his explanation of the Fourier transform, and I was wondering how it is actually computed.
To compute the center of the mass we may use: $$ (\bar x,\bar y)=\frac{\int_C (x,y) d s}{\int_C d s}\quad\text{with}\quad ds=\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\;dt. $$
Applying this to your particular curve with $ds=\left|\cos\frac t4\right|dt$ one obtains: $$ (\bar x,\bar y)=\frac{\int_0^{4\pi} (\cos\frac t2+\frac12\cos t,\sin\frac t2+\frac12\sin t) \left|\cos\frac t4\right|dt}{\int_0^{4\pi}\left|\cos\frac t4\right|dt }=\left(0.3,0\right). $$