Final answer:
The volume of the solid bounded by the paraboloid z = 7x2 + 4y2 and the planes x = 0, y = 2, y = x, z = 0 is found by setting up a double integral over the triangular region of integration and evaluating it.
Step-by-step explanation:
To find the volume of the solid enclosed by the paraboloid z = 7x2 + 4y2 and the planes x = 0, y = 2, y = x, and z = 0, we need to set up an integral. The given boundaries describe a region in the xy-plane that is a triangular slice with vertices at (0,0), (2,2), and (0,2). The volume can be found by integrating the function that defines the upper surface, z, across this region.
Step by Step Solution:
- Sketch the region of integration in the xy-plane.
- Set the limits of integration for x from 0 to 2, and for y from x to 2.
- Write down the double integral: ∫∫ (7x2 + 4y2) dy dx.
- Evaluate the integral to find the volume.
To calculate the volume precisely, we would integrate z from the xy-plane to the surface of the paraboloid, yielding the formula for the volume: V = ∫∫ (7x2 + 4y2) dy dx, evaluating as mentioned between the provided limits. However, without the exact integration process, we can make no conclusion regarding factors like π or numbers such as 7 in relation to the cube's side r.