$ $$\int_1^2 \int_{1/y}^{y} \ \sqrt[]{\frac yx} e^\sqrt[]{\ x y } dx\,dy $
In the above integral, for evaluating it, I have to change to new variables $ u={\sqrt xy } $ and v=$ \sqrt[]{\frac yx}$ . But after that how can I identify limits for new variables u and v. I can draw the region in x-y plane. But how can I find the limit here in these kind of substitutions, is there any general method? Please help
The answer is "u" varies from 1 to 2 and "v" varies from 1 to 2/u. How?
$$x=\dfrac{u}{v}\\y=uv$$therefore $$\dfrac{1}{y}<x<y\\1<y<2$$leads to $$\dfrac{1}{uv}<\dfrac{u}{v}<uv\\1<uv<2$$the first inequality leads to $$1<u,1<v$$and the second one yields to $$\dfrac{1}{u}<v<\dfrac{2}{u}$$which means that $$1<u<2\\1<v<\dfrac{2}{u}$$