Question

Find the volumes described by the following:

0 ≤ R ≤ 5; 0 ≤ θ ≤ π/3; 0 ≤ ϕ ≤ 2π

Answer 1

To find the volume described by given ranges of cylindrical coordinates, integrate the volume element over the given ranges.

Step-by-step explanation:

The given problem requires finding the volume of a solid described by certain values of the cylindrical coordinates R, θ, and ϕ. To find the volume, we need to integrate the volume element dV over the given ranges of R, θ, and ϕ. The volume element in cylindrical coordinates is given by dV = R dR dθ dz.

Therefore, the volume can be calculated as:

V = ∫02π ∫0π/3 ∫05 R dR dθ dz

Performing the integration, the volume is found to be 125π/2 cubic units.