Use the Divergence Theorem to evaluate $ \iint_{S} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathrm{N}} d s $ Verify your answer by evaluating the integral as a triple integral.
$F ( x , y , z ) = 2 x \hat { \mathbf { i } } - 2 y \hat { \mathbf { j } } + z ^ { 2 } \hat { \mathbf { k } }$
$S$ : cube bounded by the planes $x = 0 , x = 2 , y = 0 , y = 2 , z = 0 , z = 2$
A) 16
B) 4
C) 32
D) 8
E) 64

Question

Use the Divergence Theorem to evaluate $ \iint_{S} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathrm{N}} d s $ Verify your answer by evaluating the integral as a triple integral. 
$F ( x , y , z ) = 2 x \hat { \mathbf { i } } - 2 y \hat { \mathbf { j } } + z ^ { 2 } \hat { \mathbf { k } }$
$S$ : cube bounded by the planes $x = 0 , x = 2 , y = 0 , y = 2 , z = 0 , z = 2$
A) 16
B) 4
C) 32
D) 8
E) 64

Quizplus · Accepted Answer

Using the Divergence Theorem, we have$$\iint_{S} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathrm{N}} d s = \iiint_{E} 
abla \cdot \overrightarrow{\mathbf{F}} d V = \iiint_{E} (2 + 2 + 2z) d V = \int_{0}^{2} \int_{0}^{2} \int_{0}^{2} 6 d z d y d x = 16$$

Use the Divergence Theorem to Evaluate ∬SF→⋅N→ds \iint_{S} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathrm{N}} d s ∬S​F⋅Nds

Use the Divergence Theorem to Evaluate $\iint_{S} \overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathrm{N}} d s$