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

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

Use the Change of Variables x=2u−23vx = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } vx=2​u−32​​v

Use the Change of Variables $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v$