Moment of inertia by double integration

Question

A homogeneous rectangular lamina has constant area density ρ. Find the moment of inertia of the lamina about one corner. (By double integration)

Answer 1

Moment inertia of element dI = ρ(x² + y²)dxdy. We take origin (0,0) at this vertex. Let say oposite vertex at (a,b). Then moment inertia of rectangle is: I = ∫∫(R)ρ(x² + y²)dxdy = ∫₀^a∫ρdx∫₀^b(x² + y²)dy = ρ∫₀^a(x²y + y³/3)₀^bdx = ρ∫₀^a(x²b + b³/3)dx = ρ(x³b/3 + b³x/3)₀^a = ρ(a³b/3 + b³a/3) = ρab(a² + b²)/3;

Answer: Moment Inertia of rectangle about it vertex I = ρab(a² + b²)/3;