Question

1) Express the triple integral ∭e^f(x, y, z) dV as an iterated integral for the given function f and solid region E.

2) Evaluate the iterated integral.
a) ( ∫a^b ∫c^d ∫g(x, y)^h(x, y) e^f(x, y, z) ,dz ,dy ,dx )
b) ( ∫c^d ∫a^b ∫g(x, y)^h(x, y) e^f(x, y, z) ,dz ,dy ,dx )
c) ( ∫a^b ∫c^d ∫h(x, y)^g(x, y) e^f(x, y, z) ,dz ,dy ,dx )
d) ( ∫c^d ∫a^b ∫h(x, y)^g(x, y) e^f(x, y, z) ,dz ,dy ,dx )

Answer 1

To express the triple integral in terms of an iterated integral, select the correct order of integration and limits based on the given functions and region. Integrating in the specified order for each variable will evaluate the integral over the solid region E.

The correct answer is a) ( ∫a^b ∫c^d ∫g(x, y)^h(x, y) e^f(x, y, z) ,dz ,dy ,dx )

Step-by-step explanation:

To express the triple integral ∫∫∫e^f(x, y, z) dV as an iterated integral for the given function f and solid region E, we can convert it into one of the given iterated integral forms. The correct form of the iterated integral depends on the order in which the integration is performed and the limits of integration for each variable.

Here, we assume the limits for x are from a to b, for y are from c to d, and for z, the limits are functions of x and y, specifically from g(x, y) to h(x, y).

The options a), b), c), and d) represent different orderings of the dx, dy, and dz integrations. To evaluate the iterated integral, we'd follow the specified order of integration.

To evaluate, the general steps would be to:

1. Insert the limits of integration for z, then integrate with respect to z.
2. Continue with the integration over y, inserting the limits for y and integrating.
3. Finally, integrate with respect to x, using the limits from a to b.

This process would yield the value of the triple integral over the specified solid region E.

The correct answer is a) ( ∫a^b ∫c^d ∫g(x, y)^h(x, y) e^f(x, y, z) ,dz ,dy ,dx )