Find the mass of a disk with radius R, thickness H, and density 𝜌(𝑟, 𝜃, 𝑧) = (z/r) cos 𝜃/4.

Question

need help with homework please

Answer 1

Triple Integral Method:

Mass = ∫_{(0 to 2π)}∫_{(0 to R)}∫_{(0 to H)} (z/r)cos (θ/4) dV or ∫_{(0 to 2π)}∫_{(0 to R)}∫_{(0 to H)} ((z/r)cos (θ/4) • r)dz dr dθ.

Rewrite as Mass = ∫_{(0 to 2π)}∫_{(0 to R)} [cos (θ/4)z²/2_{(0 to H)}] dr dθ or ∫_{(0 to 2π)}∫_{(0 to R)} [cos (θ/4)H²/2] dr dθ.

Rewrite as Mass = ∫_{(0 to 2π)} cos (θ/4)H²r/2 _{(0 to R)} dθ or ∫_{(0 to 2π)} cos (θ/4)H²R/2 dθ.

Finally, rewrite as Mass = 2H²Rsin (θ/4)_{(0 to 2π)} or 2RH².