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:

Three Decimal Places

It refers to numerical values rounded or precise to the thousandths place, displaying three digits to the right of the decimal point.

\(6^{\sqrt{2}}\)

An example of a number raised to an irrational exponent, here specifically the square root of 2, showcasing an exponential expression.

\(4e^x\)

An exponential function where the base is Euler's number (e) multiplied by 4, and \(x\) is the exponent.

Simplify

The process of altering an expression to make it easier to understand or work with, often by reducing it to its simplest form.