Solve the Problem $$\iiint \left( 1 - \frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 64 } - \frac { z ^ { 2 } } { 64 } \right) d V ?$$

question 221

Multiple Choice

Solve the problem.
-What domain D in space minimizes the value of the integral
$$\iiint \left( 1 - \frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 64 } - \frac { z ^ { 2 } } { 64 } \right) d V ?$$

• A) \(\mathrm { D } =\) the boundary and interior of the sphere \(x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64\).
• B) \(D =\) the boundary of the sphere \(x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64\).
• C) \(\mathrm { D } = 23\)
• D) No such maximum domain exists.

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Definitions:

Demographer

A social scientist who studies population dynamics, including birth rates, mortality rates, and migration patterns.

Net Migration Rate

The difference between the number of immigrants and the number of emigrants throughout the year, per 1,000 inhabitants in a given country.

Emigrants

Individuals who leave their own country to settle permanently in another.

Immigrants

Individuals who move from one country to another with the intention of settling permanently or temporarily in the new country.