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 integral is approximated by evaluating the function at the centers of the cube's faces, which are $(\pm 1, 0, 0)$, $(0, \pm 1, 0)$, and $(0, 0, \pm 1)$. The function values are $\sqrt{1^2 + 2 \cdot 0^2 + 7 \cdot 0^2} = 1$, $\sqrt{0^2 + 2 \cdot 1^2 + 7 \cdot 0^2} = \sqrt{2}$, and $\sqrt{0^2 + 2 \cdot 0^2 + 7 \cdot 1^2} = \sqrt{7}$. Each face contributes equally, and there are 6 faces, so the sum is $6(1 + \sqrt{2} + \sqrt{7})$, which simplifies to $8(1 + \sqrt{2} + \sqrt{7})$.

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

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