Final answer:
The volume of the solid under the paraboloid z = x² + y² and inside the cylinder x² + y² = 2x is π/2. Integration in polar coordinates is used, with r ranging from 0 to 1 and θ from 0 to 2π.
Step-by-step explanation:
To find the volume of the solid that lies under the paraboloid z = x² + y², above the xy-plane, and inside the cylinder x² + y ²= 2x, we need to set up a double integral. First, we complete the square for the cylinder equation to get (x - 1)² + y² = 1, which represents a cylinder of radius 1 centered at (1,0). We use polar coordinates for the integration where x = r²cos(θ) and y = r²sin(θ). The limits of integration for r will be from 0 to 1 (the radius of the cylinder), and for θ, from 0 to 2π to cover the entire circle. The volume integral in polar coordinates is then ∫∫ z r dr dθ.
To evaluate the integral, we substitute the equation of the paraboloid, getting the integral ∫∫ (r²) r dr dθ, which simplifies to ∫ (0 to 2π) dθ ∫ (0 to 1) r³ dr. The result of the radial integral is r´/4 evaluated from 0 to 1, and the angular integral is simply 2π. Therefore, the volume is (1/4)2π, which equals π/2.