Evaluate $\oint _ { C } \left( 12 x ^ { 2 } y ^ { 2 } - 6 y z ^ { 2 } \right) d x + \left( 8 x ^ { 3 } y - 6 x z ^ { 2 } \right) d y - 12 x y z d z$

question 66

Short Answer

Evaluate $\oint _ { C } \left( 12 x ^ { 2 } y ^ { 2 } - 6 y z ^ { 2 } \right) d x + \left( 8 x ^ { 3 } y - 6 x z ^ { 2 } \right) d y - 12 x y z d z$ , where the path C is the curve of intersection of the paraboloid $3 z = x ^ { 2 } + y ^ { 2 }$ with the plane $3 x = 4 y + z = 12$ .

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Definitions:

External Costs

Costs incurred by society as a whole due to economic activities but not reflected in the market price or borne by the producer or consumer involved.

Pollution Abatement

Actions or measures taken to reduce, control, or eliminate pollution and its negative impacts on the environment.

Socially Optimal

A condition or output level where the social welfare is maximized, taking into account both the benefits to consumers and the costs to producers.

Marginal Benefit

The additional satisfaction or utility that an individual gains from consuming one more unit of a good or service.