University:
University of Cambridge
Course:
0580 IGCSE | Mathematics
Academic year:

2023
Views:

473

Pages:

3
Author:

Shahbaz A.

Ex 15.7 Page 884 Q23 THOMAS’ CALCULUS twelfth Edition shahbaz ahmed July 2024 Spherical Coordinates In spherical coordinates position of a point is described by ρ, r, θ as in figure below 1 Q23 Evaluate the spherical coordinate integral Z 2π Z 0 0 π Z ( 1−cos ϕ) 2 ρ2 sin ϕ dρ dϕ dθ 0 Solution 2π Z 0 2π 2π π Z 0 π 0 0 ρ2 sin ϕ dρ dϕ dθ = ϕ) ( 1−cos 2 0 Z Z 2π Z 0 Z 0 Z ϕ) ( 1−cos 2 Z 0 0 Z Z π Z ϕ) ( 1−cos 2 0 1 ρ2 sin ϕ dρ dϕ dθ = 3 0 2π Z 1 ρ2 sin ϕ dρ dϕ dθ = 24 # ρ2 dρ sin ϕ dϕ dθ 0 Z 0 π ϕ 3 1−cos ρ 0 2 sin ϕ dϕ dθ 0 2π Z ϕ) ( 1−cos 2 "Z π 0 Z π (1 − cos ϕ)3 sin ϕ dϕ dθ 0 d(1 − cos ϕ) = sin ϕ dϕ Z 0 2π Z 0 π Z 0 ϕ) ( 1−cos 2 1 ρ2 sin ϕ dρ dϕ dθ = 24 2 2π Z 0 Z 0 π (1 − cos ϕ)3 d(1 − cos ϕ) dθ 2π Z 0 Z π ϕ) ( 1−cos 2 Z 0 0 π (1 − cos ϕ)4 0 π (1 − cos ϕ)4 0 π (1 − cos ϕ)4 0 π (1 − cos ϕ)4 0 π (1 − cos ϕ)4 0 2π Z π 2π ϕ) ( 1−cos 2 Z 0 Z π Z 0 0 Z 2π 2π 0 π (1 − cos ϕ)4 0 dθ = 16 0 Z Z = (1 − cos π)4 − (1 − cos 0)4 = [1 − (−1)]4 − (1 − 1)4 = [1 + 1]4 = [2]4 Z 0 1 ρ2 sin ϕ dρ dϕ dθ = 96 Z Z 0 2π ϕ) ( 1−cos 2 π Z ϕ) ( 1−cos 2 0 π Z dθ 0 1 [θ]2π 0 6 ρ2 sin ϕ dρ dϕ dθ = 1 (2π) 6 0 Z 2π Z ρ2 sin ϕ dρ dϕ dθ = 0 0 0 16 ρ sin ϕ dρ dϕ dθ = 96 2 ϕ) ( 1−cos 2 ρ2 sin ϕ dρ dϕ dθ = 0 3 π 3