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:

Marginal Social Benefit

The supplementary gain for society derived from the consumption of one extra unit of a good or service.

Marginal Social Cost

The cost to society of producing an additional unit of a good or service, including both private costs and any externalities.

Marginal Social Benefit

The additional gain to society as a whole from an additional unit of a good or service.

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A tax imposed on the emission of pollutants, aimed at reducing the environmental externality costs of these emissions.