Find the coordinates of the center of the mass of the curve $$ x^2+y^2=1, x+2y+3z=12 $$
I find calculating line integrals in 3D problematic and really don't know how to approach this one. I think that the curve we get is an elipse but how to find its parameters?
Parameterise the circle $x^2+y^2=1$ as
$$(x,y) = (\cos(t),\sin(t)) \qquad t \in [0,2\pi]$$
as usual. Plug this into the other equation:
$$x+2y+3z=12 \implies \cos(t)+2\sin(t)+3z=12 \implies z = 4 - \frac{\cos(t)+2\sin(t)}{3}$$
So the parameterisation is
$$(x,y,z) = \bigg(\cos(t),\sin(t),4 - \frac{\cos(t)+2\sin(t)}{3}\bigg) \qquad t \in [0,2\pi]$$