Example 5 Evaluate S F n d s where F 12 x2 y (2024)

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Example 5 : Evaluate $\iint_{S} \bar{F}, \bar{n} d s$ where $\bar{F}=12 x^{2} y \bar{i}-3 y \bar{j}+2 z \bar{k}$ and $\mathrm{S}$ is the portion of he plane $x+y+z=1$ included in the first octant.

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#### Solution By Steps***Step 1: Find the Outward Unit Normal Vector $\bar{n}$***The equation of the plane is $x+y+z=1$. The coefficients of $x$, $y$, and $z$ in this equation give the components of the normal vector $\bar{n}$. Therefore, $\bar{n}=\left(\begin{array}{lll} 1 & 1 & 1 \end{array}\right)$.***Step 2: Normalize the Normal Vector $\bar{n}$***To find the unit normal vector $\bar{n}$, divide $\bar{n}$ by its magnitude. The magnitude of $\bar{n}$ is $\left\| \bar{n} \right\|=\sqrt{1^{2}+1^{2}+1^{2}}=\sqrt{3}$. Therefore, the unit normal vector is $\bar{n}=\left(\begin{array}{lll} \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} \end{array}\right)$.***Step 3: Calculate the Dot Product $\bar{F} \cdot \bar{n}$***The dot product of $\bar{F}$ and $\bar{n}$ is $\bar{F} \cdot \bar{n}=12x^{2}y\left(\frac{1}{\sqrt{3}}\right)-3y\left(\frac{1}{\sqrt{3}}\right)+2z\left(\frac{1}{\sqrt{3}}\right)$.***Step 4: Integrate over the Region***Integrate the dot product $\bar{F} \cdot \bar{n}$ over the region $S$ in the first octant defined by $x+y+z=1$.#### Final AnswerThe value of $\iint_{S} \bar{F} \cdot \bar{n} ds$ over the region $S$ in the first octant is $\frac{1}{\sqrt{3}}\int_{0}^{1}\int_{0}^{1-x} (12x^{2}y-3y+2z)dydz$. This integral can be computed to find the final result.

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