Let E Be the Solid That Lies Below the Sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$

question 95

Multiple Choice

Let E be the solid that lies below the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ and above the cone $\phi = \beta$ , where $0 < \beta < \frac { \pi } { 2 }$ . Find the value of the triple integral $\iiint _ { E } z d V$ .

• A) \(\pi a ^ { 2 } \sin \beta\)
• B). \(\frac { 1 } { 2 } \pi a ^ { 2 } \sin \beta\)
• C) \(\pi a ^ { 2 } \sin ^ { 2 } \beta\)
• D) \(\frac { 1 } { 2 } \pi a ^ { 2 } \sin ^ { 2 } \beta\)
• E) \(\frac { 1 } { 4 } \pi a ^ { 2 } \sin ^ { 2 } \beta\)
• F) \(\frac { 1 } { 2 } \pi \alpha ^ { 4 } \sin ^ { 2 } \beta\)
• G) \(\frac { 1 } { 4 } \pi a ^ { 4 } \sin ^ { 2 } \beta\)

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Definitions:

Manual Recording

The process of documenting information by hand, as opposed to using electronic means.

Debit Sheet

A record listing all debit transactions, often used in accounting to monitor expenditures.

Double-Entry Bookkeeping System

A method of bookkeeping where every entry to an account requires a corresponding and opposite entry to a different account, ensuring the accounting equation balances.

Single-Entry System

An accounting method where each financial transaction is recorded with a single entry to the accounting records, less complex than double-entry accounting.