Find the Surface Area of the Surface S $S$ Is the Portion of the Paraboloid

question 246

Multiple Choice

Find the surface area of the surface S.
- $S$ is the portion of the paraboloid $z = 81 - x ^ { 2 } - y ^ { 2 }$ that lies above the ring $16 \leq x ^ { 2 } + y ^ { 2 } \leq 36$ in the $x y$ -plane.

• A) \(\frac { \pi } { 4 } [ 145 \sqrt { 145 } - 65 \sqrt { 65 } ]\)
• B) \(\frac { \pi } { 2 } [ 145 \sqrt { 145 } - 65 \sqrt { 65 } ]\)
• C) \(\frac { \pi } { 6 } [ 145 \sqrt { 145 } - 65 \sqrt { 65 } ]\)
• D) \(\frac { \pi } { 3 } [ 145 \sqrt { 145 } - 65 \sqrt { 65 } ]\)

---

Definitions:

Epimers

Stereoisomers that differ in configuration at exactly one stereogenic center and have the same molecular formula.

Asymmetric Carbons

Carbon atoms bonded to four different groups, leading to chirality or "handedness" in molecules, essential in stereochemistry.

α-D-Ribopyranose

An aldopentose sugar in the pyranose form where the α-configuration at the anomeric carbon is specific to the D-ribose structure.

Ordinal Data

A type of categorical statistical data type in which the order of the values is significant but the difference between each one is not necessarily uniform.