Final answer:
The student's question involves converting a triple integral from Cartesian to cylindrical coordinates and evaluating it. The region of integration is a cylinder within the bounds of the radius 3 and height determined by the function 9 - r^2. The process includes setting up the bounds for r, θ, and z appropriately in cylindrical coordinates and integrating step by step.
Step-by-step explanation:
The student is asking to evaluate a triple integral of a function in Cartesian coordinates and convert it to cylindrical coordinates. The given region is a cylinder with a circular base in the xy-plane and extending along the z-axis. Using cylindrical coordinates, we express the Cartesian coordinates (x, y, z) in terms of (r, θ, z), where r is the radial distance, θ is the angular coordinate, and z is the same as in Cartesian coordinates. For this example, x2 + y2 = r2, and the bounds for r will go from 0 to 3 (since the given circular region has a radius of 3), θ will range from 0 to 2π (completing a full circle around the z-axis), and z will be bounded from 0 to 9 - r2.
Therefore, the integral in cylindrical coordinates will be:
∫02πdθ ∫03rdr ∫09-r2√(r2)dz
This simplifies the process of integration since we are now dealing with simpler integrals of r and θ. Performing the integrals step by step should yield the required result.