Find the Area of the Surface $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u ^ { 2 } \mathbf { k }$

question 96

Multiple Choice

Find the area of the surface. The part of the paraboloid $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u ^ { 2 } \mathbf { k }$ ; $0 \leq u \leq 2$ , $0 \leq v \leq 2 \pi$

• A) \(\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 6 }\)
• B) \(\frac { 5 \pi ( 17 \sqrt { 17 } - 1 ) } { 12 }\)
• C) \(\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 12 }\)
• D) \(\frac { \pi ( 17 \sqrt { 17 } - 1 ) } { 3 }\)

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Definitions:

Facilitator of Succession

An organism or factor that catalyzes ecological succession, ensuring the progression from one ecological community to another, leading to increased biodiversity.

Sexual Maturity

The stage in an organism's life cycle when it becomes capable of reproduction.

Vulnerable

A state of being susceptible to harm or damage, often used to describe species at risk of extinction or ecosystems facing threats.