With Cartesian graphs, I understand that the graphs under the $x$-axis will cause a decrease in net area. But is the same true for parametics?
Say a curve C is defined by the $x = 3t^2, y = 2t^3$. There's a tangent at $t=2$ which also intersects $C$ at $t=-1$. Do I integrate from $-1$ to $2$?
Yes, that is so when you are summing up trapeziums making up the area. However when reckoning a tangent at $t=2,$ the summing up modality changes. Just proceed as suggested by its parameterization, area is positive and automatically takes care of sign on either side of $x$ axis. It is as if $t$ travels along the curve.
$$ Area= \int y\, dx = \int y\, \frac{dx}{dt} dt $$
$$ = \int_{-1}^2 2t^3\, 6t\, dt = \int_{-1}^2 12 t^4 dt = \frac{12}{5} (2^5-(-1)^5) = 12\cdot 33/5$$