Find the Volume of the Solid Generated by Revolving the Region

question 33

Multiple Choice

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x \text {-axis. }$ -axis. Verify your results using the integration capabilities of a Graphing utility.
$y = \sin ( x ) , y = 0 , x = 0 , x = \frac { \pi } { 3 }$

• A) \(\frac { 1 } { 3 } \pi ^ { 2 } - \frac { \sqrt { 3 } } { 8 } \pi\)
• B)\(\frac { 1 } { 6 } \pi ^ { 2 } + \frac { \sqrt { 3 } } { 8 } \pi\)
• C)\(\frac { 1 } { 6 } \pi ^ { 2 } - \frac { \sqrt { 3 } } { 8 } \pi\)
• D)\(\frac { 1 } { 3 } \pi ^ { 2 } + \frac { \sqrt { 3 } } { 8 } \pi\)
• E)\(\frac { 1 } { 6 } \pi ^ { 2 } - \frac { \sqrt { 3 } } { 4 } \pi\)

---

Definitions:

Destroyed

To damage something so badly that it no longer exists or cannot be repaired.

Impossibility of Performance

A legal defense used in contract law where a party claims that fulfilling the obligations of a contract has become objectively impossible due to unforeseen events.

Supervening Events

Events that occur after the formation of a contract which could render the performance impossible or significantly change the nature of the original agreement.

F.O.B. Buffalo

An Incoterm meaning Free On Board Buffalo, indicating that the seller delivers when the goods are loaded on board a vessel at the Buffalo location, with the seller covering transportation to that point but the buyer taking responsibility thereafter.