Find the Value of the Line Integral $\int_{c} \mathbf{F} \cdot d \mathbf{r}$

question 67

Multiple Choice

Find the value of the line integral $\int_{c} \mathbf{F} \cdot d \mathbf{r}$ , where $\mathbf{F}(x, y) =12 x y \mathbf{i}+6 x^{2} \mathbf{j}$ and $\mathbf { r } ( t ) = 8 t \mathbf { i } + 8 t ^ { 2 } \mathbf { j } , 0 \leq t \leq 1$
(Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)

• A) 3,072
• B) 192
• C) 6,144
• D) 6,144.
• E) 6,912.

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Definitions:

Beck

Aaron T. Beck, a psychiatrist known for developing cognitive therapy, particularly for depression.

Overgeneralizing

A cognitive distortion involving broad, sweeping conclusions based on a single event or piece of evidence.

Selective Abstraction

A cognitive distortion where an individual focuses on a single negative detail or event, often ignoring the broader context of a situation.

Dichotomous Thinking

A cognitive distortion involving seeing things in black and white categories or all or nothing terms, without recognizing the middle ground.