Final answer:
To calculate the volume of the solid defined by a parabolic cylinder and planes using a triple integral, set up an integral with the variable limits for x taken from the cylinder's equation and the given bounds for y and z.
Step-by-step explanation:
The student is asking to use a triple integral to find the volume of a solid. This solid is bounded by the parabolic cylinder y = 4x2, and the planes z = 0, z = 9, and y = 6. To find the volume using a triple integral, the integrand would be simply 1, since we are integrating over a volume in three-dimensional space. The limits of integration for x are found by solving for x from the equation y = 4x2. Since y is bounded by 6, that implies ±√(6/4) or ±√(3/2) are the bounds for x. For y, the bounds are 0 to 6 because of the cylinder reaching only up to y = 6. And the bounds for z are from 0 to 9 based on the provided planes. The triple integral thus becomes ∫∫∫ dz dy dx, evaluated with limits z from 0 to 9, y from 0 to 6, and x from ±√(3/2).