Evaluate the Line Integral $\int _ { C } \left( 2 x y ^ { 2 } + \tan ^ { - 1 } x ^ { 3 } \right) d x + \left( x ^ { 2 } y + \sqrt { y ^ { 3 } + 1 } \right) d y$

question 217

Short Answer

Evaluate the line integral $\int _ { C } \left( 2 x y ^ { 2 } + \tan ^ { - 1 } x ^ { 3 } \right) d x + \left( x ^ { 2 } y + \sqrt { y ^ { 3 } + 1 } \right) d y$ , where C consists of the arc of the parabola $y = x ^ { 2 }$ from $( 1,1 )$ to $( 2,4 )$ , followed by the line segments from $( 2,4 )$ to $( 0,4 )$ , from $( 0,4 )$ to $( 0,1 )$ and from $( 0,1 )$ to $( 1,1 )$ .

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Definitions:

Cis-1-T-Butyl-4-Methylcyclohexane

A specific isomer of a chemical compound characterized by the cis configuration of 1-tert-butyl and 4-methyl groups on a cyclohexane ring.

Stable Conformation

A specific spatial arrangement of the atoms in a molecule that is energetically favorable and has minimal strain.

Equilibrium Constants

Represents the ratio of concentrations of products to reactants at equilibrium, specific to a particular chemical reaction, denoted as K_eq.

Equilibrium

A state in a chemical reaction where the rates of the forward and reverse reactions are equal, leading to no net change in the concentration of reactants and products over time.