Find a Function $f$ Such That $\mathbf { F } = \nabla f$

question 104

Short Answer

Find a function $f$ such that $\mathbf { F } = \nabla f$ , and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the given curve $C$ .
$$\begin{array} { l } \mathbf { F } ( x , y ) = x ^ { 3 } y ^ { 4 } \mathbf { i } + y ^ { 3 } x ^ { 4 } \mathbf { j } \\C : \mathbf { r } ( t ) = \sqrt { t } \mathbf { i } + \left( 1 + t ^ { 3 } \right) \mathbf { j } , 0 \leq t \leq 1\end{array}$$

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Definitions:

Amphetamine

A potent central nervous system stimulant used medicinally in the treatment of attention deficit hyperactivity disorder (ADHD), narcolepsy, and obesity, but also misused illegally for its energetic and euphoric effects.

Central Nervous System

The central nervous system consists of the brain and spinal cord, functioning as the primary processing center for the body's nervous system.

Stimulant

A substance that raises levels of physiological or nervous activity in the body, typically used to increase alertness, energy, or physical performance.

Tranquilizer

A drug used to reduce tension or anxiety, often by depressing nerve function.