Use the Transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$

Use the transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ to evaluate the integral $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$ , where $R$ is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 2$ .

A) $\frac { 100 \pi } { 3 ^ { 3 / 2 } }$
B) $\frac { 25 } { \sqrt { 3 } }$
C) $\frac { 5 \pi } { \sqrt { 3 } }$
D) $\frac { 5 \pi ^ { 2 } } { \sqrt { 3 } }$
E) $\frac { 4 \pi } { \sqrt { 3 } }$

Correct Answer:

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