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Pgfind the volume of the solid enclosed by the paraboloidsz=9x2 y2andz=89x2 y2.

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Final answer:

To find the volume of the solid enclosed by the given paraboloids, set up a triple integral in cylindrical coordinates and perform the integration within the appropriate limits.

Step-by-step explanation:

The question requires finding the volume of a solid enclosed by two paraboloids. The paraboloids in question are described by the equations z=9x2+y2 and z=8-9x2-y2. Integrating to find the volume can be achieved by setting up a triple integral in cylindrical coordinates for efficiency due to the symmetry of the paraboloids. The limits of integration for r (the radial coordinate) are from 0 to a, where a is the radius where the two paraboloids intersect. The limits for θ (the angular coordinate) are from 0 to 2π, spanning the entire circle. Lastly, the limits for z are from the bottom paraboloid (z=9r2) to the top one (z=8-9r2). By performing the integration, one can obtain the exact volume of the solid.

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