Final answer:
To calculate the triple integral in cylindrical coordinates, express the function in terms of cylindrical coordinates and define the limits of integration for each variable. Perform the integral in the specified order, and evaluate the result.
Step-by-step explanation:
To calculate the triple integral of a given function in cylindrical coordinates, we need to express the function in terms of the cylindrical coordinates (ρ, θ, z), and define the limits of integration for each variable. Let's say the function is f(ρ, θ, z) = ρ². The region of integration is a cylindrical volume bounded by ρ = 2 and ρ = 5, θ varies from 0 to 2π, and z varies from -∞ to +∞.
The integral becomes: ∫∫∫ f(ρ, θ, z) ρ dρ dθ dz, where the limits of integration are: 0 to 2π for θ, -∞ to +∞ for z, and 2 to 5 for ρ. Note that the integrals can be computed in any order, but for simplicity, we can start with the innermost integral with respect to ρ.
After integrating ρ² with respect to ρ from 2 to 5, we get (1/3)(5³ - 2³) = (1/3)(125 - 8) = 119/3.