Find the Exact Area of the Surface Obtained by Rotating $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

question 13

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:

Substitution Effect

The change in consumption patterns due to a change in relative prices, where consumers replace more expensive items with less costly alternatives.

Inferior Good

Items for which demand falls when the income levels of buyers go up, unlike standard goods that see an increase in demand with rising incomes.

Income Effect

How shifts in the financial status of individuals or whole economies affect their purchasing desires for specific goods or services.

Price Increase

An increase in the price of products or services.