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

Question

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

Quizplus · Accepted Answer

The correct answer is G.The solid E is a cone with radius a and height a sin(beta). The volume of the cone is given by V = (1/3)πa^2 sin(beta). The integral of z over the cone is given by ∫∫∫E z dV = ∫0^2π ∫0^a sin(beta) ∫0^a sin(phi) z dz dphi dr = (1/4)πa^4 sin^2(beta).

Let E Be the Solid That Lies Below the Sphere x2+y2+z2=a2x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }x2+y2+z2=a2

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