Do you think my professor just screwed up when writing this calculus problem, or am I overlooking something?

Yeah, your professor screwed up. The graphs do not intersect.

Man, I had not made graphs in a long time.

I’m guessing the prof meant y^2=x+4 not y^2=x-4.

@bob_ Good graph! I didn’t even try to plot it, when I saw yours.

The lamina is bounded by two equations, which do not intersect, does not mean that the lamina does not exist (in maths though). It is just unbounded. Boundaries of the lamina are defined by those equations; and boundaries stretch out to infinity.

If that question comes in exam, I would answer infinity for mass; as lamina is unbounded or limitless; so, its mass would be limitless or infinity. Mass of lamina is summation of masses of all particles or points on the lamina.
Centroid calculations are troublesome though. Had density been uniform, centroid & center of mass would have been same; and centroid would be simply where the lamina would be found symmetrical. But, density varies with y.

So, if this happens to come in exam, I would have skipped this part of question!!! Or I would have just blindly answered somewhere where the lamina looks more or less symmetrical (if there is no negative marking)!