Chapter 13.3, Problem 69E

A mass calculation Suppose the density of a thin plate represented by the region R is ρ(r, θ) (in units of mass per area). The mass of the plate is ${\displaystyle {\iint }_R\rho \left(r,\theta \right)}dA$. Find the mass of the thin half annulus R = {(r, θ): 1 ≤ r ≤ 4,0 ≤ θ ≤ π} with a density ρ(r, θ) = 4 + r sin θ.