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

2023
Views:

115

Pages:

2
Author:

Shahbaz A.

Ex 15.7 Page 884 Q27 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 Q27 Evaluate the spherical coordinate integral 2 Z Z 0 ρ3 sin 2ϕ dϕ dθ dρ π 4 −π 0 π 2 Z Solution 2 Z 0 Z π 2 Z 2 Z ρ3 sin 2ϕ 0 Z 2 Z π 2 Z ρ3 sin 2ϕ "Z 0 # π 2 sin 2ϕ dϕ ρ3 dθ dρ = dϕ dθ dρ = − π 4 1 2 dθ dρ π 4 −π 0 −π 0 dϕ π 4 −π 0 Z Z 2 0 Z π [cos 2ϕ] π2 ρ3 dθ dρ 4 −π 0 π [cos 2ϕ] π2 = cos 2( π2 ) − cos 2( π4 ) = [cos π − cos π2 ] = −1 4 Z 0 2 0 Z 2 ρ3 sin 2ϕ dϕ π 4 −π Z π 2 Z Z 0 π 2 Z ρ Z 2 sin 2ϕ dϕ π 4 −π 0 3 1 dθ dρ = − × (−1) 2 0 Z 0 π 2 Z −π π 4 2 0 Z Z ρ3 sin 2ϕ π 2 Z ρ 2 Z −π 0 Z 0 Z 2 Z 0 π 2 Z 2 Z 0 −π dθ dρ = dϕ ρ3 sin 2ϕ dϕ Z 2 π 2 Z π 2 0 Z 0 dθ 1 2 2 Z 0 0 Z π 2 Z 2 ρ3 dρ 0 2 ρ3 dρ 0 ρ3 sin 2ϕ dϕ dθ dρ = π 1 42 × [ρ ]0 2 4 ρ3 sin 2ϕ dϕ dθ dρ = π [16] = 2π 8 π 4 2 ρ3 dρ [ θ]−π ρ3 dρ π dθ dρ = 2 dθ dρ = ρ3 dρ −π 0 −π π 4 Z Z dθ 0 π 4 −π 0 Z 0 sin 2ϕ dϕ π 4 −π 0 3 1 dθ dρ = 2 2 Z