Evaluate the Surface Integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$

question 117

Short Answer

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ for the vector field $\mathbf { F } ( x , y , z ) = x ^ { 2 } y \mathbf { i } - 3 x y ^ { 2 } \mathbf { j } + 4 y ^ { 3 } \mathbf { k }$ where S is the part of the elliptic paraboloid $z = x ^ { 2 } + y ^ { 2 } - 9$ that lies below the square $0 \leq x \leq 1 , \quad 0 \leq y \leq 1$ and has downward orientation.

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Definitions:

Marginal Cost

It refers to the cost change associated with producing an extra unit of a good or service, important in economic decision-making.

Fixed Cost

Costs that do not vary with the level of output, such as rent or salaries.

Variable Costs

Expenses that directly fluctuate in relation to the volume of production or output.

Profit-maximizing Output

The peak production point for a firm where it attains its greatest possible profit, characterized by the equality of marginal cost and marginal revenue.