Evaluate integral in spherical coordinate
Zπ Zπ 2sinϕ
Z
ρ2 sin ϕ dρ dθ dϕ
0 0 0
Solution.R R R
π π 2sinϕ 2
Let I = 0 0 0 ρ sin ϕ dρ dθ dϕ
R π R π R 2sinϕ 2
I=[ 0 0[ 0 ρ dϕ] sin ϕ dρ dθ ]
RπRπ
I = 31 [ 0 0 [ρ3 ]2i=0
sin ϕ
sin ϕ dϕ dθ]
RπRπ
I = 13 [ 0 0 [(2 sin ϕ)3 − 03 ] sin ϕ dϕ dθ]
RπRπ
I = 13 [ 0 0 [8 sin3 ϕ] sin ϕ dϕ dθ]
RπRπ
I = 83 [ 0 0 [sin3 ϕ] sin ϕ dϕ dθ]
RπRπ
I = 83 [ 0 0 [sin4 ϕ dϕ] dθ]
1
Zπ Zπ 2sinϕ
Z
ρ2 sin ϕ dρ dθ dϕ
0 0 0
Solution.R R R
π π 2sinϕ 2
Let I = 0 0 0 ρ sin ϕ dρ dθ dϕ
R π R π R 2sinϕ 2
I=[ 0 0[ 0 ρ dϕ] sin ϕ dρ dθ ]
RπRπ
I = 31 [ 0 0 [ρ3 ]2i=0
sin ϕ
sin ϕ dϕ dθ]
RπRπ
I = 13 [ 0 0 [(2 sin ϕ)3 − 03 ] sin ϕ dϕ dθ]
RπRπ
I = 13 [ 0 0 [8 sin3 ϕ] sin ϕ dϕ dθ]
RπRπ
I = 83 [ 0 0 [sin3 ϕ] sin ϕ dϕ dθ]
RπRπ
I = 83 [ 0 0 [sin4 ϕ dϕ] dθ]
1
