Solve the Problem $z$ -Axis Shell: "Nose" of the Paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$

question 64

Multiple Choice

Solve the problem.
-The shape and density of a thin shell are indicated below. Find the moment of inertia about the $z$ -axis. Shell: "nose" of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ cut by the plane $z = 2$
Density: $\delta = \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + 1 } }$

• A) \(I _ { Z } = \frac { 64 } { 3 } \pi\)
• B) \(\mathrm { I } _ { \mathrm { Z } } = 128 \pi\)
• C) \(\mathrm { I } _ { Z } = \frac { 16 } { 3 } \pi\)
• D) \(I _ { Z } = 8 \pi\)

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Definitions:

Cerebral Cortex

The cerebral cortex is the outer layer of the brain, involved in high-level brain functions such as thought, language, and consciousness.

Corpus Callosum

The corpus callosum is a wide band of nerve fibers connecting the left and right hemispheres of the brain, facilitating interhemispheric communication.

Communicate

The act of conveying information, feelings, or thoughts from one individual or group to another through various means, including verbal, non-verbal, or written channels.

Corpus Callosum

A thick band of nerve fibers that divides the cerebral cortex lobes into left and right hemispheres, facilitating communication between the two halves.