Let $\vec { F } = x \vec { j }$

question 48

Essay

Let $\vec { F } = x \vec { j }$ Let C₁ be the line from (0, 0)to (2, 0), C₂ the line from (2, 0)to (2,-1), C₃ the line from (2,-1)to (0,-1), and C₄ the line from (0,-1)to (0, 0).
(A)Using the definition of line integral only, without parameterizing the curves, show that the line integral of $\vec{F}$ along C = C₁ + C₂ + C₃ + C₄ is -2.That is, show $\int_{C} \vec{F} \cdot \overline{d r}=-2$ (B)The rectangle, R, enclosed by the lines C₁, C₂, C₃ and C₄ is of area 2.So, by Green's Theorem $\int _ { C } \vec { F } \cdot \overline { d r } = \int _ { R } \left( \frac { \partial x } { \partial x } - 0 \right) d A = \text { Area of } R = 2$ Is something wrong?

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Definitions:

Sleepwalks

The act of walking or performing other complex behaviors while in a state of sleep.

REM Sleep

A phase of sleep characterized by rapid eye movements, where vivid dreaming often occurs, and considered important for processing emotions and memories.

Slow-Wave Sleep

A deep sleep stage characterized by slow brain waves, where recovery and growth tend to happen.

Conscious Awareness

The state of being awake and aware of one’s surroundings, thoughts, and experiences.