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

2023
Views:

25

Pages:

3
Author:

Shahbaz A.

Q22 Spherical Coordinates shahbaz ahmed July 2024 Spherical Coordinates In spherical coordinates position of a point is described by ρ, r, θ as in figure below 1 Ex 15.7 Page 884 Q22 THOMAS’ CALCULUS twelfth Edition Q22 R 2π R 0 π 4 R2 0 ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ 0 Solution Z 2π π 4 Z 2 Z Z 2 2π π 4 Z 2 Z ρ cos ϕ ρ sin ϕ dρ dϕ dθ = 0 0 0 2π Z π 4 Z 0 ρ 0 2 Z 0 1 ρ cos ϕ ρ sin ϕ dρ dϕ dθ = 4 2 0 dϕ dθ 0 0 2π Z dρ cos ϕ sin ϕ 3 π 4 Z 0 0 4 2 ρ 0 cos ϕ sin ϕ dϕ dθ Now [ρ4 ]20 = 16 Z 2π π 4 Z 0 2 Z 0 16 ρ cos ϕ ρ sin ϕ dρ dϕ dθ = 4 2 0 Z 2π "Z # π 4 cos ϕ sin ϕ 0 dϕ dθ 0 d(cos ϕ) = − sin ϕ dϕ −d(cos ϕ) = sin ϕ dϕ Z 2π π 4 Z 2 Z Z 2 2π "Z ρ cos ϕ ρ sin ϕ dρ dϕ dθ = −4 0 0 0 2π Z Z 2π π 4 π 4 Z 0 2 Z 0 0 cos ϕ d(cos ϕ) 0 Z ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = − 0 2 Z Z 2 4 2 0 Z 2π π 4 Z 0 0 0 Z 0 2π ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = −2 0 Z 0 Z h i π cos2 − cos2 0 dθ 4 2π h i π cos2 − cos2 0 dθ 4 0 π 4 Z 2 ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = −2 0 0 Z 0 2 2 π4 cos ϕ 0 dθ 2π 0 2 Z 2π Z ρ cos ϕ ρ sin ϕ dρ dϕ dθ = −2 0 # π 4 2π 1 − 1 dθ 2 dθ Z 2π Z 0 π 4 0 0 Z 0 2π 2 Z Z 0 π 4 1 ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = (−2)(− ) 2 Z Z 2π dθ 0 2 ρ cos ϕ ρ2 sin ϕ dρ dϕ dθ = [θ]2π 0 = 2π 0 3