Find the average value of the function f (x, y, z) = 5x2z 5y2z over the region enclosed by the paraboloid z = 4 - x2 - y2 and the plane z = 0.

The paraboloid meets the x-y plane when x²+y²=9. A circle of radius 3, centre origin.

Use cylindrical coordinates (r,θ, z) so paraboloid becomes z = 9-r² and f = 5r²z.

If F is the mean of f over the region R then F ∫ (R) dV = ∫ (R) fdV

∫ (R) dV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9-r²] rdrdθdz

= ∫∫ [θ=0,2π, r=0,3] r (9-r²) drdθ = ∫ [θ=0,2π] { (9/2) 3² - (1/4) 3⁴} dθ = 81π/2

∫ (R) fdV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9-r²] 5r²z. rdrdθdz

= 5∫∫ [θ=0,2π, r=0,3] ½r³{ (9-r²) ² - 0 } drdθ

= (5/2) ∫∫ [θ=0,2π, r=0,3] { 81r³ - 18r⁵ + r⁷} drdθ

= (5/2) ∫ [θ=0,2π] { (81/4) 3⁴ - (3) 3⁶ + (1/8) 3⁸} dθ = 10935π/8

∴ F = 10935π/8 : 81π/2 = 135/4