Final Answer:
The surface area of the part of the circular paraboloid z = x² + y² that lies inside the cylinder x² + y² = 9 is A = 27π square units.
Step-by-step explanation:
To find the surface area of the circular paraboloid inside the cylinder, we use the surface area formula for a surface given by z = f(x, y), which is given by A = ∬_D √(1 + (∂f/∂x)² + (∂f/∂y)²) dA. For the circular paraboloid z = x² + y², the partial derivatives are (∂f/∂x) = 2x and (∂f/∂y) = 2y.
The region D is the projection of the surface onto the xy-plane, which is the circular region inside the cylinder given by x² + y² ≤ 9. Using polar coordinates, 0 ≤ r ≤ 3 and 0 ≤ θ ≤ 2π, the surface area integral becomes A = ∫₀^(2π) ∫₀³ √(1 + (2rcosθ)² + (2rsinθ)²) r dr dθ.
Solving this double integral yields A = 27π square units, representing the surface area of the circular paraboloid inside the cylinder. The process involves careful consideration of the geometry and parameterization of the given surfaces, as well as the use of polar coordinates to simplify the integration over the circular region inside the cylinder