Plane region, uniform: A = ∫(top − bot) dx, My = ∫x(top − bot) dx, Mx = ½∫(top² − bot²) dx, then x̄ = My/A and ȳ = Mx/A.
The squared one gives the coordinate you are not integrating along. Integrating in y instead swaps which formula gives which — the usual place to slip.
With density ρ(x): put ρ inside every integral. A constant ρ cancels out of x̄ and ȳ but not out of the mass.
Arc (a wire): weight by ds, not by area — x̄ = (1/L)∫x ds. A straight wire's centroid is its midpoint.
Pappus: V = 2πd·A and S = 2πd·L, with d the distance from the centroid to the axis. Works in both directions — a known volume hands you a centroid.
Polar: the Jacobian is r, so A = ∬r dr dθ and My = ∬r²cos θ dr dθ.
Second moments weight by the square of the distance: Ix = ∬y²ρ dA, Iy = ∬x²ρ dA, and always I0 = Ix + Iy.
Worth knowing cold: triangle ⅓ · simplex in n-D 1/(n+1) · parabolic segment ⅖ of the height · half-disc 4R/3π · semicircular wire 2R/π · solid hemisphere 3R/8 · cone h/4.
Answers: a centroid is a pair — 2/3, 1/3. Single-value problems take one number: 9/5, 32pi^2, 4/(3pi), or a decimal.