Find the Exact Area of the Surface Obtained by Rotating $x$

question 20

Multiple Choice

Find the exact area of the surface obtained by rotating the given curve about the $x$ -axis.
$$x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$$

• A) \(\frac { 24 \pi } { 5 }\)
• B) \(\frac { 18 \pi } { 5 }\)
• C) \(\frac { 12 \pi } { 5 }\)
• D) \(\frac { 2 \pi } { 4 }\)
• E) None of these

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Definitions:

Respiratory Membrane

The respiratory membrane is the barrier through which gas exchange occurs between the air in the lungs and the blood in the capillaries.

Diffusion

The movement of molecules from an area of higher concentration to an area of lower concentration, a fundamental process in physics and biology.

Alveolar Ventilation Rate

The rate at which air moves in and out of the alveoli, the tiny air sacs in the lungs, crucial for gas exchange.

Minute Respiratory Volume

The total volume of air breathed in and out by a person at rest in one minute, important in assessing respiratory health.