Evaluate $\iint _ { S } f ( x , y , z ) d S$

question 49

Multiple Choice

Evaluate $\iint _ { S } f ( x , y , z ) d S$ .
$f ( x , y , z ) = z ; S$ is the part of the torus with vector representation
$\mathbf { r } ( u , v ) = ( 5 + 3 \cos v ) \cos u \mathbf { i } + ( 5 + 3 \cos v ) \sin u \mathbf { j } + 3 \sin v \mathbf { k } , 0 \leq u \leq 2 \pi , 0 \leq v \leq \frac { \pi } { 2 }$

• A) 0
• B) \(117 \pi\)
• C) 117
• D) \(27 \pi\)

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Definitions:

Cannon-Bard Theory

A psychological theory proposing that emotions and physical reactions, such as heart rate and sweating, occur simultaneously and independently in response to a stimulus.

Physiological Changes

Alterations in the function or structure of living organisms that occur naturally over time or due to various factors.

Fear

An emotional response to perceived threats or danger, characterized by feelings of anxiety, tension, and increased alertness.

James-Lange Theory

A theory of emotion proposing that emotions occur as a result of physiological responses to external stimuli.