Solve the Problem $$\iiint _ { R } \int \sqrt { 1 - \left( \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } + \frac { z ^ { 2 } } { 9 } \right) } d x d y d z$$

question 259

Multiple Choice

Solve the problem.
-Evaluate
$$\iiint _ { R } \int \sqrt { 1 - \left( \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } + \frac { z ^ { 2 } } { 9 } \right) } d x d y d z$$
where $R$ is the interior of the ellipsoid $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } + \frac { z ^ { 2 } } { 9 } = 1$ . Hint: Let $x = 5 u , y = 4 v , z = 3 w$ , and then convert to spherical coordinates.

• A) \(15 \pi\)
• B) \(20 \pi\)
• C) \(20 \pi ^ { 2 }\)
• D) \(15 \pi ^ { 2 }\)

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Definitions:

Amino Acids

Fundamental components of proteins that include a basic amino group, an acidic carboxyl group, and a distinct side chain.

Glucose-6-Phosphate

A glucose sugar phosphorylated at the 6 position, a critical step in glucose metabolism and glycogen synthesis.

Glucose

Glucose is a simple sugar that serves as a primary source of energy for the body's cells, obtained from consumed carbohydrates and required for metabolic processes.

Metabolize

The process by which living organisms convert substances into energy and new materials, supporting growth, reproduction, and maintenance of life.