Final answer:
To find the area of the cap cut from the paraboloid by the cone, you need to determine the region of intersection between the two shapes and then evaluate a double integral.
Step-by-step explanation:
To find the area of the cap cut from the paraboloid by the cone, we need to determine the region where the paraboloid and the cone intersect. The equation of the paraboloid is given as z = 2 - x² - y², and the equation of the cone is z = px² y². We can set the two equations equal to each other to find the intersection points. By substituting the equation of the cone into the equation of the paraboloid, we get 2 - x² - y² = px² y². Rearranging the equation, we have x² + py²x²y² + y² = 2. This equation represents the region of intersection between the paraboloid and the cone.
To find the area of the cap, we need to integrate the area element over the region of intersection. The area element is given by dA = √(1 + (dz/dx)² + (dz/dy)²) dxdy. The limits for the integral will depend on the region of intersection, which can be found by solving the equation x² + py²x²y² + y² = 2 for x and y. Once we have the limits, we can evaluate the double integral to find the area of the cap.