Find the Value of the Line Integral $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } + 8 \mathbf { k } , 0 \leq t \leq \pi$

question 33

Multiple Choice

Find the value of the line integral $\int_{c} \mathbf { F } \text {. } d \mathbf { r } \text {, where } \mathbf { F } ( x , y ) = e ^ { y } ( z \mathbf { i } + x z \mathbf { j } + x \mathbf { k } ) \text { and }$ $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } + 8 \mathbf { k } , 0 \leq t \leq \pi$
(Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)

• A) - 24
• B) 48
• C) - 48
• D) 24
• E) 0

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Consumption

The act of using goods and services for personal needs or to generate utility.

Utility Function

A mathematical representation of how different combinations of goods or services can result in varying levels of satisfaction or utility to a consumer.

Consumer's Surplus

The difference between what consumers are willing to pay for a good or service and what they actually pay, representing the benefit consumers receive from the purchase.

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Small devices inserted into the ear canal to protect the ears from loud noises, water, or foreign bodies.