While solving double integrals for finding area between curves, how to decide the transformations.
Eg. In this integral $\int_0^1 \int_0^x dy dx$ we use $u=x+y$ and $v=x-y$.
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In the case of the example why would I use the suggested transformation? I can do the integration: $\int_0^1 \int_0^x dy dx=\int_0^1x\ dx=\frac12.$ so I don't understand the question. (There are many ways to transform the coordinates. In this case I (we) did not need any tranformatin.) – zoli Nov 30 '17 at 00:50
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@zoli in this example it wasn't necessary, but for easier understanding I took this problem. Questions like https://math.stackexchange.com/questions/680953/change-of-variables-in-double-integral-explanation-of-solved-example?rq=1 are better solved by transformation. – Vincent Nov 30 '17 at 00:54
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Beware of this particular transformation, it changes the domain of integration to rotated rectangles and this can lead to pitfalls. See this related question for instance : https://math.stackexchange.com/questions/2133856/can-indefinite-double-integrals-be-solved-by-change-of-variables-technique – zwim Nov 30 '17 at 01:35