Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$

question 222

Multiple Choice

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( e ^ { x ^ { 2 } } - y ^ { 3 } , \cos y ^ { 2 } + x ^ { 3 } \right)$ and the curve C is given by C : $x = 2 \sin t , y = 2 \cos t , 0 \leq t \leq 2 \pi$ .

• A) \(24 \pi\)
• B)\(12 \pi\)
• C)\(48 \pi\)
• D) \(96 \pi\)
• E) \(- 24 \pi\)
• F)\(- 12 \pi\)
• G) \(- 48 \pi\)
• H)\(0\)

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Definitions:

Dimethylamine

Dimethylamine is a simple amine consisting of two methyl groups attached to a nitrogen atom, used in a wide range of chemical applications, including as a precursor for certain chemicals.

3-pentanone

An organic compound characterized by a five-carbon chain with a ketone functional group at the third carbon atom.

Enamine

Derivatives of alkenes formed from the reaction between a secondary amine and a carbonyl group, often used as intermediates in organic synthesis.

Imine

A functional group or compound featuring a carbon-nitrogen double bond, often formed by the condensation of an aldehyde or ketone with an amine.