Evaluate the surface integral of the function g over the surface S.
-$G ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ; S$ is the surface of the cube formed from the coordinate planes and the planes $x = 1$, $y = 1$, and $z = 1$
A) 5
B) $\frac { 5 } { 3 }$
C) $\frac { 7 } { 3 }$
D) 7

Question

Quizplus · Accepted Answer

We can divide the surface of the cube into six rectangular sections (i.e. the top, bottom, front, back, left, and right sides). Notice that on each section, two of the variables are constant while the third variable varies over the range [0,1]. We can then set up the surface integral by computing the integral over each of these sections and adding them up.

For example, consider the integral over the front side of the cube (i.e. where $z = 0$). The surface is a rectangle with vertices (0,0,0), (1,0,0), (1,1,0), and (0,1,0). The normal vector to the surface is (0,0,-1). Therefore, the surface integral over this section is given by:

\begin{aligned} \iint _ { S } g \cdot d \mathbf { S } & = \iint _ { \text {front side} } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) \cdot \mathbf { n } \ d S \ & = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \left( x ^ { 2 } + y ^ { 2 } \right) \cdot (0,0,-1) \ d x \ d y \ & = \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } 0 \ d x \ d y \ & = 0 \end{aligned}

Similarly, we can compute the surface integral over each of the other five sections and add them up. Doing so, we get:

$$\iint _ { S } g \cdot d \mathbf { S } = 6 \cdot \left( 2 \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } x ^ { 2 } + y ^ { 2 } \ d x \ d y + \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } z ^ { 2 } \ d x \ d y \right)$$

Evaluating the integrals, we get:

\begin{aligned} \iint _ { S } g \cdot d \mathbf { S } & = 6 \cdot \left( \int _ { 0 } ^ { 1 } \left[ x ^ { 3 } / 3 + xy ^ { 2 } / 2 \right] _ { 0 } ^ { 1 } dy + \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } z ^ { 2 } \ d x \ d y \right) \ & = 6 \cdot \left( \int _ { 0 } ^ { 1 } \left( 1 / 3 + y ^ { 2 } / 2 \right) \ dy + \int _ { 0 } ^ { 1 } \frac{1}{3} \ dy \right) \ & = 6 \cdot \left( \frac{5}{6} \right) \ & = 7 \end{aligned}

Therefore, the answer is D) 7.

Evaluate the Surface Integral of the Function G Over the Surface