Final answer:
To find the volume of the solid bounded by the given paraboloids, we convert the equation to cylindrical coordinates and set up a triple integral with the appropriate limits. The integration is performed first with respect to y, then r, and finally θ to obtain the volume of the solid.
Step-by-step explanation:
To find the volume of the solid enclosed by the paraboloids y = x² + z² and y = 50 − x² − z², we will use a triple integral. The two paraboloids intersect where y = x² + z² equals y = 50 − x² − z², which simplifies to x² + z² = 25. This is the equation of a circle in the xz-plane with radius 5. To set up our triple integral, we change to cylindrical coordinates (r, θ, y) since the region of integration is circular.
The limits of integration for r will be from 0 to 5, the limits for θ will be from 0 to 2π (a full circle), and the limits for y will range from the bottom surface, r², to the top surface, 50 − r². The triple integral becomes:
∫∫∫ (50 − 2r²) r dr dθ dy
To evaluate, integrate first with respect to y, then r, and finally θ. The integral with respect to y just involves computing the difference between the top and bottom functions, which is (50 − r²) − r² = 50 − 2r². Integrating this with respect to r gives us the integral of r(50 - 2r²) from 0 to 5, and integrating the result with respect to θ over the interval from 0 to 2π gives us the volume of the solid.