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

2023
Views:

246

Pages:

3
Author:

Shahbaz A.

Ex 15.7 Page 884 Q25 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 Q25 Evaluate the spherical coordinate integral 2π Z 0 3ρ2 sin ϕ dρ dϕ dθ = 3ρ 0 2π Z 0 Z 2 3ρ 0 2π Z 0 2π Z π 3 Z π 3 Z 3ρ2 sin ϕ dρ dϕ dθ = 0 2π π 3 0 Z 0 0 π 3 π 3 π 3 0 Z 0 π 3 Z "Z 2π "Z 2π dθ − 0 3ρ2 sin ϕ dρ dϕ dθ = 4[θ]2π 0 − 2 dθ # tan ϕ d(tan ϕ) 2 sec ϕ dϕ dθ 0 0 Z 2 π 3 dθ − [cos ϕ]0 dϕ dθ # π 3 sec ϕ tan ϕ Z π 3 sec3 ϕ sin ϕ 0 0 1 3ρ2 sin ϕ dρ dϕ dθ = (−8)(− ) 2 sec ϕ 2π dϕ dθ 0 2π Z 2 Z π 3 Z 0 0 Z 2π Z 2π dϕ dθ (8 − sec3 ϕ) sin ϕ 0 Z dϕ dθ 0 sin ϕ dϕ dθ − 3ρ2 sin ϕ dρ dϕ dθ = −8 Z Z 0 sec ϕ 2π 2π 2 Z dρ sin ϕ 3 2 ρ sec ϕ sin ϕ 0 sin ϕ dρ dϕ dθ = 8 sec ϕ Z 0 Z 2 8 sin ϕ dϕ dθ − Z 2 2π π 3 π 3 Z 0 3ρ Z 2π Z Z 0 0 2 Z 2π Z sin ϕ dρ dϕ dθ = sec ϕ 0 0 Z 2 Z 0 2 sec ϕ 0 2 sec ϕ 0 3 3 3ρ2 sin ϕ dρ dϕ dθ = sec ϕ π 3 Z Z ρ 0 2 Z 0 2π Z π 3 Z 3ρ2 sin ϕ dρ dϕ dθ sec ϕ π 3 Z 2 sin ϕ dρ dϕ dθ = 3 sec ϕ 0 Z 0 2π Z 2 π 3 Z 0 2 Z 2π Z sec ϕ π 3 Z 2 Z 0 2π Z π 3 Z 1 2 Z 1 ×3 2 2π 2 tan ϕ 0 Z 2π dθ 0 π3 0 dθ Z 2π Z 0 π 3 Z 2π 0 Z 0 π 3 Z Z 0 2 Z sec ϕ 0 2π 3ρ2 sin ϕ dρ dϕ dθ = 4[2π] − sec ϕ 0 Z 2 π 3 Z 1 × 3[θ]2π 0 2 3 3ρ2 sin ϕ dρ dϕ dθ = 4[2π] − [2π] 2 2 3ρ2 sin ϕ dρ dϕ dθ = 8π − 3π = 5π sec ϕ 3