Find the volume of the solid enclosed by the paraboloids z = 16(x^2+y^2) and z = 2-16(x^2-y^2))

Answer 1

Find the intersection: The paraboloids intersect where:

1. 16(x² + y²) = 2 - 16(x² + y²)
2. 32(x² + y²) = 2
3. x² + y² = 1/16
4. This is a circle of radius r = 1/4

Set up the volume integral: Using cylindrical coordinates (r, θ, z):

1. V = ∫∫ (z_upper - z_lower) dA
2. V = ∫₀^(2π) ∫₀^(1/4) [2 - 32r²] r dr dθ
3. Evaluate:∫₀^(1/4) (2r - 32r³) dr = [r² - 8r⁴]₀^(1/4) = 1/16 - 1/32 = 1/32
4. ∫₀^(2π) (1/32) dθ = (1/32)(2π) = π/16