Final answer:
The triple integral can be expressed in six ways as an iterated integral, based on the six possible integration orders of x, y, and z, with appropriate limits for each variable.
Step-by-step explanation:
The question asks us to express the triple integral ∫∫∫_E f(x,y,z) dV over a solid E as an iterated integral in six different ways. The solid E is bounded by the surfaces z=y-8x and y=24. The six different iterated integral representations correspond to the six possible orders of integrating with respect to x, y, and z.
For illustration, one way to set up the iterated integral is by fixing the order of integration: first x, then y, and lastly z. The integration limits for x can be expressed in terms of y and z from the equation z = y - 8x, and the integration limits for y are given by the constraint y = 24. The order in which the variables are integrated can be changed, resulting in different iterated integrals. Each change in order may require a reevaluation of the integration limits based on the relationships between x, y, and z.