Question

Use spherical coordinates x=rcos(ψ)y=rsin(ψ)cos(θ) and z=rsin(ϕ)sin(θ) where 0≤θ<2π and 0≤ψ≤π, to find the volume of the 3 -dimensional ball B³ᵣ = {(x,y,z) ∈ R³ ∣ x²+y²+z² ≤ r² }

Answer 1

To find the volume of the 3-dimensional ball B³ᵣ, we can use the formula for the volume of a sphere in spherical coordinates.

Step-by-step explanation:

To find the volume of the 3-dimensional ball B³ᵣ = {(x,y,z) ∈ R³ ∣ x²+y²+z² ≤ r² } using spherical coordinates x=rcos(ψ)y=rsin(ψ)cos(θ) and z=rsin(ϕ)sin(θ), we can use the formula for the volume of a sphere in spherical coordinates.

The volume of a sphere in spherical coordinates is given by:

V = ∫∫∫ r²sin(ϕ) dr dϕ dθ

As the limits of integration for r, ϕ, and θ are 0 to r, 0 to π, and 0 to 2π respectively, the volume can be calculated as:

V = ∫0r∫0π∫02π r²sin(ϕ) dθ dϕ dr