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

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) $\frac { 1 } { 4 } \pi a ^ { 4 } \sin ^ { 2 } \beta$

H) $\pi a ^ { 4 } \sin \beta$

Correct Answer:

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