Set Up a Triple Integral That Gives the Moment of Inertia

question 110

Multiple Choice

Set up a triple integral that gives the moment of inertia about the $z \text {-axis of the solid }$ region $Q$ of density given below.
$\begin{array} { l } \rho ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \\Q = \{ - 4 \leq x \leq 4 , - 4 \leq y \leq 4,0 \leq z \leq 1 - y \}\end{array}$

• A)
• \( \int_{-4}^{4}\int_{-4}^{4}\int_{0}^{1-x}\left(x^{2}+y^{2}\right) \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x \)
• B)
• \( \int_{-4}^{4}\int_{-4}^{4}\int_{0}^{1-x}\left(x^{2}+y^{2}\right) \sqrt{x^{2}+y^{2}+z^{2}} d y d z d x \)
• C)
• \( \int_{-4}^{4}\int_{-4}^{4}\int_{0}^{1-y}\left(x^{2}+y^{2}\right) \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x \)
• D)
• \( \int_{-4}^{4}\int_{-4}^{4}\int_{0}^{1-x}\left(x^{2}+z^{2}\right) \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x \)
• E)
• \( \int_{-4}^{4}\int_{-4}^{4}\int_{0}^{1-y}\left(z^{2}+y^{2}\right) \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x \)

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Definitions:

Diminishing Returns

Diminishing returns, a concept in economics, occurs when adding an additional factor of production results in a lower increment of output, assuming other factors are constant.

Total Cost

The entire expense of production, incorporating both consistent and fluctuating costs.

Produce

Refers to agricultural products, especially fresh fruits and vegetables, created through farming.

Utility

Utility is a measure of satisfaction or happiness that consumers derive from consuming goods and services.