Evaluate the Surface Integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$

question 155

Multiple Choice

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = x \mathbf { i } + y \mathbf { j } + x \mathbf { k }$ and S is the part of the plane $z = 1 - x - y$ in the first octant with downward orientation.

• A) \(\frac { 1 } { 3 }\)
• B) \(\frac { 1 } { 2 }\)
• C) \(\frac { 1 } { 4 }\)
• D)\(1\)
• E) \(- \frac { 1 } { 3 }\)
• F)\(- \frac { 1 } { 2 }\)
• G)\(- \frac { 1 } { 4 }\)
• H)\(- 1\)

---

Definitions:

Healthiest Way

Refers to the optimal approach or method to maintain or improve health, often involving nutrition, physical activity, and mental well-being practices.

Bulimia Nervosa

An eating disorder characterized by cycles of binge eating followed by purging, to prevent weight gain.

Binge Eat

The consumption of large quantities of food in a short period, often as a response to emotional stress, beyond the point of feeling comfortably full.

Excessive Exercise

Engaging in physical activity to a degree that exceeds recommended levels, potentially leading to negative health outcomes.