Let $S$ Be the Cube with Vertices $( \pm 1 , \pm 1 , \pm 1 )$

Multiple Choice

Let $S$ be the cube with vertices $( \pm 1 , \pm 1 , \pm 1 )$ . Approximate $\iint _ { S } \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 7 z ^ { 2 } }$ by using a
Riemann sum as in Definition 1, taking the patches $S _ { i j }$ to be the squares that are the faces of the cube and the points $P _ { i j }$ to be the centers of the squares.

A) $8 ( 3 + \sqrt { 2 } )$
B) $8 ( 3 + \sqrt { 7 } )$
C) $8 ( 1 + \sqrt { 2 } + \sqrt { 7 } )$
D) $4 ( 1 + \sqrt { 2 } + \sqrt { 7 } )$
E) none of these