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

The paraboloid intersects the x-y plane when x²+y²=9, defining a circle with a radius of 3, centered at the origin.

<span>Utilizing cylindrical coordinates (r,θ,z), the paraboloid transforms into z = 9−r² and f = 5r²z. </span>

<span>If F represents the average of f over the area R then F ∫ (R)dV = ∫ (R)fdV </span>

<span>∫ (R)dV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9−r²] rdrdθdz </span>

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


<span>∫ (R)fdV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9−r²] 5r²z.rdrdθdz </span>

<span>= 5∫∫ [θ=0,2π, r=0,3] ½r³{ (9−r²)² − 0 } drdθ </span>

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

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

<span>Thus, F = 10935π/8 ÷ 81π/2 = 135/4</span>