Chapter 15.8, Problem 51E

(a) Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere r² + z² = a² and below by the cone z = r cot ϕ₀ (or ϕ = ϕ₀), where 0 < ϕ₀ < π/2, is V = $\frac{2\pi a^{3.}}{3}$ (1 – cosϕ₀)

(b) Deduce that the volume of the spherical wedge given by ρ₁ $\le$ ρ $\le$ ρ₂, ${\theta }_1$ $\le$ $\theta$ $\le$ ${\theta }_2$_, ϕ₁ $\le$ ϕ $\le$ ϕ₂ is ΔV = $\frac{{\rho }_2{}^3- {\rho }_1{}^3}{3}$(cosϕ_{1 –} cos ϕ₂)( ${\theta }_2$ – ${\theta }_1$)

(c) Use the Mean Value Theorem to show that the volume in part (b) can be written as ΔV = ${\rho }^{-2}$sin $\bar{\varphi }$Δρ Δ $\theta$ Δϕ where $\bar{\rho }$ lies between ρ₁ and ρ₂, $\bar{\varphi }$ lies between ϕ₁ and ϕ₂, Δρ = ρ₂ – ρ₁, Δ= $\theta$₂ – $\theta$₂, and Δ = ϕ₂ – ϕ₁.