The Surface of the Dome on a New Museum Is $\mathbf { r } = ( u , v ) = 24 \sin u \cos v \mathbf { i } + 24 \sin u \sin v \mathbf { j } + 24 \cos u \mathbf { k }$

Multiple Choice

The surface of the dome on a new museum is given by $\mathbf { r } = ( u , v ) = 24 \sin u \cos v \mathbf { i } + 24 \sin u \sin v \mathbf { j } + 24 \cos u \mathbf { k }$ , where $0 \leq u \leq \frac { \pi } { 3 }$ and $0 \leq v \leq 2 \pi$ and $\mathbf { r }$ is in meters. Find the surface area of the dome.

A) $288 \pi \mathrm { m } ^ { 2 }$
B) $1152 \pi \mathrm { m } ^ { 2 }$
C) $24 \pi \mathrm { m } ^ { 2 }$
D) $576 \pi \mathrm { m } ^ { 2 }$
E) $48 \pi \mathrm { m } ^ { 2 }$