Let R be the region bounded between the plane z = 1 and the cone
z=2-sqrt(x^2+y^2)
. Describe
region R using spherical coordinate
To describe the region R using spherical coordinates, we need to express the equations of the plane and the cone in spherical coordinates.
In spherical coordinates, we typically use the variables (ρ, θ, φ), where ρ represents the distance from the origin to the point, θ represents the azimuthal angle measured from the positive x-axis in the xy-plane, and φ represents the inclination angle measured from the positive z-axis.
First, let's express the equation of the plane z = 1 in spherical coordinates. Since the plane z = 1 is a constant value, we can directly say that ρ = 1, regardless of the values of θ and φ.
Next, let's express the equation of the cone z = 2 - √(x^2 + y^2) in spherical coordinates. We can rewrite this equation in terms of ρ, θ, and φ using the following relationships:
x = ρsinφcosθ
y = ρsinφsinθ
z= ρcosφ
Substituting these values into the equation of the cone, we get:
ρcosφ = 2 - √(ρ^2sin^2φcos^2θ + ρ^2sin^2φsin^2θ)
ρcosφ = 2 - √(ρ^2sin^2φcos^2θ + ρ^2sin^2φsin^2θ)
ρcosφ = 2 - √(ρ^2sin^2φ(cos^2θ+sin^2θ))
ρcosφ = 2 - √(ρ^2sin^2φ)
Now, let's solve for ρ in terms of the spherical coordinates:
ρcosφ = 2 - ρsinφ
ρ(sinφ + cosφ) = 2
ρ = 2 / (sinφ + cosφ)
So, in spherical coordinates, the equation of the cone becomes:
ρ = 2 / (sinφ + cosφ)
Now, we can describe the region R in spherical coordinates as follows:
ρ = 2 / (sinφ + cosφ), with 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π, and 0 ≤ ρ ≤ 1.
This represents the region bounded between the plane z = 1 and the cone z = 2 - √(x^2 + y^2) in spherical coordinates.