Evaluate $\iint _ { S } 4 x y d S$, where $S$ is $z = 5 - x - y$, first octant.
A) $\frac { 125 \sqrt { 2 } } { 2 }$
B) $\frac { 4,625 \sqrt { 2 } } { 2 }$
C) $\frac { 4,625 \sqrt { 3 } } { 2 }$
D) $\frac { 625 \sqrt { 3 } } { 6 }$

Question

Evaluate $\iint _ { S } 4 x y d S$, where $S$ is $z = 5 - x - y$, first octant. 
A) $\frac { 125 \sqrt { 2 } } { 2 }$
B) $\frac { 4,625 \sqrt { 2 } } { 2 }$
C) $\frac { 4,625 \sqrt { 3 } } { 2 }$
D) $\frac { 625 \sqrt { 3 } } { 6 }$

Quizplus · Accepted Answer

First, we need to find the limits of integration. Since the plane is in the first octant, we have:
$$0 \leq x \leq 5$$
$$0 \leq y \leq 5-x$$
And since the plane goes up to $z=5-x-y$, we have:
$$0 \leq z \leq 5-x-y$$
Now we can set up the integral:
$$\iint _ { S } 4 x y d S = \int _ { 0 } ^ { 5 } \int _ { 0 } ^ { 5-x } \int _ { 0 } ^ { 5-x-y } 4xy dzdydx$$
Solving the integral, we get:
$$\frac { 625 \sqrt { 3 } } { 6 }$$

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