Evaluate the Line Integral Over the Given Curve $C$ $$\int _ { C } \left( 6 x + 2 y ^ { 2 } \right) d s ; C : \mathbf { r } ( t ) = ( t - 6 ) \mathbf { i } + t \mathbf { j } , 0 \leq t \leq 1$$

question 37

Multiple Choice

Evaluate the line integral over the given curve $C$ .
$$\int _ { C } \left( 6 x + 2 y ^ { 2 } \right) d s ; C : \mathbf { r } ( t ) = ( t - 6 ) \mathbf { i } + t \mathbf { j } , 0 \leq t \leq 1$$

• A) 28
• B) \(- 26\)
• C) \(40 \sqrt { 2 }\)
• D) \(- \frac { 97 } { 3 } \sqrt { 2 }\)

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Definitions:

Reduced Cost

In linear programming, the amount by which the objective function coefficient of a variable must decrease before that variable's value increases in the optimal solution.

Bounded Variable

A variable that has upper and lower limits.

Shadow Price

A monetary value assigned to currently unpriced goods or services, reflecting their worth in terms of opportunity cost or their valuation in a constrained optimization problem.

Reduced Cost

In optimization, it's the difference between the cost of a non-basic variable and its contribution to the objective function, used in linear programming.