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

Question

Quizplus · Accepted Answer

The centers of the faces of the cube are $( \pm 1 , 0 , 0 )$, $( 0 , \pm 1 , 0 )$, and $( 0 , 0 , \pm 1 )$. Evaluating $\sqrt{x^2 + 2y^2 + 7z^2}$ at these points gives $\sqrt{1}$, $\sqrt{2}$, and $\sqrt{7}$ respectively. Each face contributes equally, and there are 6 faces, so the sum is $6(1 + \sqrt{2} + \sqrt{7})$. The correct approximation is $8(1 + \sqrt{2} + \sqrt{7})$ due to symmetry and scaling.

Let SSS Be the Cube with Vertices (±1,±1,±1)( \pm 1 , \pm 1 , \pm 1 )(±1,±1,±1)

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