Show That $\mathbf { F }$ Is Conservative and Find a Function

question 87

Multiple Choice

Show that $\mathbf { F }$ is conservative and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use this result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any path from $A \left( x _ { 0 } , y _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } \right)$ .
$\mathbf { F } ( x , y ) = \left( 8 x ^ { 3 } y ^ { 2 } - 6 x y ^ { 2 } \right) \mathbf { i } + \left( 4 x ^ { 4 } y - 6 x ^ { 2 } y \right) \mathbf { j } ; A ( - 2,0 )$ and $B ( - 1,2 )$
Select the correct answer.

• A) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 60\)
• B) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 238\)
• C) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = 24\)
• D) \(\int _ { C } \mathbf { F } \cdot d \mathbf { r } = - 4\)

---

Definitions:

Neurochemical Properties

Characteristics and behaviors of the chemicals in the nervous system that influence brain function and behavior.

Nicotine

A potent chemical found in tobacco plants that acts as a stimulant and is highly addictive, commonly consumed through smoking cigarettes.

Cocaine

A powerful stimulant drug derived from the leaves of the coca plant, which can cause intense high and is highly addictive.

Amphetamines

A class of potent stimulant drugs that affect the central nervous system, enhancing alertness and energy.