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Chapter 13: Problem 2

Write an iterated integral for \(\iiint_{D} f(x, y, z) d V,\) where \(D\) is the box \(\\{(x, y, z): 0 \leq x \leq 3,0 \leq y \leq 6,0 \leq z \leq 4\\}\)

Short Answer

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Question: Write the iterated integral expression for the function f(x, y, z) within the region D, which is a box with boundaries x from 0 to 3, y from 0 to 6, and z from 0 to 4. Answer: \(\iiint_D f(x, y, z) \ dV = \int_{0}^{3} \int_{0}^{6} \int_{0}^{4} f(x, y, z) \ dz \ dy \ dx\)

Step by step solution

01

Identify the integration limits for x, y, and z

The given region D is a box with the following boundaries for x, y, and z coordinates: - 0 鈮� x 鈮� 3 - 0 鈮� y 鈮� 6 - 0 鈮� z 鈮� 4 We will use these limits when setting up our iterated integral.
02

Compose the iterated integral expression

Now, we will write the expression for the iterated integral using the identified limits in the previous step. The volume integral for the given function f(x, y, z) within the region D is given by the following expression: \(\iiint_D f(x, y, z) \ dV = \int_{0}^{3} \int_{0}^{6} \int_{0}^{4} f(x, y, z) \ dz \ dy \ dx\) This is the required iterated integral expression for the given function within the box region D, as per the given constraints.

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