Use the Given Transformation to Evaluate the Integral Where $R$ Is the Interior of the Ellipsoid

question 78

Multiple Choice

Use the given transformation to evaluate the integral.
- $$\begin{array} { l } x = 9 u , y = 10 v , z = 4 w ; \\\quad \int _ { R } \iint \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) d x d y d z\end{array}$$
where $R$ is the interior of the ellipsoid $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 16 } = 1$

• A) \(144 \pi\)
• B) \(216 \pi\)
• C) \(240 \pi\)
• D) \(288 \pi\)

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Definitions:

Substitution Effect

The alteration in purchasing behavior caused by variations in the relative costs of products, prompting buyers to opt for less expensive alternatives over pricier ones.

Price Change

An adjustment in the cost of a good or service in the market, which can be an increase or decrease due to various factors.

Income

The remuneration accrued, notably in a recurrent manner, from labor or the investment of resources.

Substitution Effect

The economic principle that as prices rise or income decreases, consumers will replace more expensive items with less costly alternatives.