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

2023
Views:

420

Pages:

3
Author:

Shahbaz A.

Ex 15.7 Page 884 Q26 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 Q26 Evaluate the spherical coordinate integral Z 2π 0 Z π 4 sec Φ Z ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ 0 0 Solution Z 2π π 4 Z sec ϕ Z ρ cos ϕ ρ 0 Z 2π 0 0 ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = 0 # sec ϕ ρ sec ϕ Z "Z π 4 Z 0 π 4 Z 2π sin ϕ dρ dϕ dθ = 0 0 Z 2 Z dρ cos ϕ sin ϕ dϕ dθ 0 0 1 4 3 2π π 4 Z 4 sec ϕ ρ 0 cos ϕ sin ϕ dϕ dθ 0 0 4 sec ϕ ρ 0 = sec4 ϕ 2π Z π 4 Z 0 0 2π Z π 4 Z Z ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = 1 4 Z π 4 sec ϕ Z 0 2π 1 4 ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = 0 0 Z sec ϕ Z 0 sec ϕ Z ρ cos ϕ ρ 0 0 Z 2π 0 Z 0 π 4 Z 0 sec ϕ 0 π 4 π [cos−2 ϕ]04 = [ √11 ( 2) π 2 − 1 cos2 0 ] − 11 ] [cos−2 ϕ]04 = [ ( 11 ) − 1] 2 [cos −2 0 sec4 ϕ cos ϕ sin ϕ dϕ dθ 0 2π Z 1 sin ϕ dρ dϕ dθ = − 4 π 4 Z π 4 Z 0 cos−3 ϕ sin ϕ dϕ dθ 0 Z 0 2π Z π 4 cos−3 ϕ d(cos ϕ) dθ 0 1 1 ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = (− ) × (− ) 4 2 π [cos−2 ϕ]04 = [ cos12 2 2π π 4 ϕ]0 = [2 − 1] = 1 2 Z 0 2π π [cos−2 ϕ]04 dθ 2π Z π 4 Z sec ϕ Z ρ cos ϕ ρ 0 0 0 2π Z 0 Z 0 2π π 4 Z 0 π 4 1 sin ϕ dρ dϕ dθ = 8 sec ϕ Z ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = 0 0 Z 2 Z sec ϕ ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = 0 3 Z 2π dθ 0 1 2π [θ] 8 0 2π π = 8 4