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

question 145

Multiple Choice

Use the given transformation to evaluate the integral.
- $$\begin{array} { l } x = 4 u , y = 5 v , z = 7 w ; \\\quad \iiint \left( \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } + \frac { z ^ { 2 } } { 49 } \right) ^ { \pi } d x d y\end{array}$$
where $R$ is the interior of the ellipsoid $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 25 } + \frac { z ^ { 2 } } { 49 } = 1$

• A) \(\frac { 280 \pi } { \pi + 1 }\)
• B) \(\frac { 560 \pi } { \pi + 2 }\)
• C) \(\frac { 560 \pi } { \pi + 3 }\)
• D) \(\frac { 560 \pi } { \pi + 1 }\)

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Definitions:

Passive Margin

The shift from oceanic to continental lithosphere which isn't located at an active plate margin.

Continental Margin

The zone of the ocean floor that separates the thin oceanic crust from thick continental crust, including the continental shelf, slope, and rise.

Plate Boundary

The edges where two tectonic plates meet, which can be divergent, convergent, or transform, leading to various geological features and events.

Normal Faults

Fractures in the Earth's crust where the rock above the fault plane has moved down relative to the rock below.