Final Answer:
The triple integral to find the volume of the solid bounded by the parabolic cylinder y = 3x² and the planes z = 0, z = 3, and y = 5 is given by: ∫₀³ ∫₀⁵ ∫₀³x² dz dy dx.
Step-by-step explanation:
To calculate the volume of the solid, we set up a triple integral using the given boundaries. The innermost integral represents the variation in the z-direction, limited by the parabolic cylinder equation y = 3x². The middle integral accounts for the y-direction, bounded by y = 5, and the outermost integral considers the x-direction within the limits 0 to 3.
Breaking down the integral, we start by integrating with respect to z from 0 to 3x², then integrate with respect to y from 0 to 5, and finally integrate with respect to x from 0 to 3. Executing these integrations step by step yields the volume of the given solid.
In essence, the triple integral evaluates the infinitesimal volume elements within the specified region, summing them up to determine the total volume of the solid. This mathematical representation allows us to compute complex spatial volumes by systematically integrating over the defined boundaries, providing a rigorous and precise approach to solving three-dimensional geometric problems.