Find the Length of the Curve $\mathrm { r } = \mathrm { e } ^ { 2 \theta } , 0 \leq \theta \leq \pi$

question 2

Multiple Choice

Find the length of the curve.
-The spiral $\mathrm { r } = \mathrm { e } ^ { 2 \theta } , 0 \leq \theta \leq \pi$

• A) \(\frac { 1 } { 8 } \left( e ^ { 4 \pi } - 1 \right)\)
• B) \(\frac { \sqrt { 5 } } { 2 } \left( e ^ { 2 \pi } - 1 \right)\)
• C) \(e ^ { 4 \pi } - \sqrt { 3 }\)
• D) \(\frac { 16 } { 3 } \left( \pi ^ { 2 } - 1 \right)\)

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Definitions:

Ecosystem

A biological community of interacting organisms and their physical environment, interconnected through nutrient cycles and energy flows.

Physical Environment

The external surroundings and conditions in which an organism lives, including land, water, air, and other natural factors influencing its growth and survival.

Living Organisms

Entities characterized by growth, reproduction, response to stimuli, and the ability to sustain metabolic processes.

Hypothesis

A proposed explanation for a phenomenon, made as a starting point for further investigation, that can be tested through experimentation and observation.