Change integration order

Question

I am confused due to graphics

$$\int_0^2\mathrm{d}x\int_{x}^{2x}f(x,y)\,\mathrm{d}y$$

well, for reverse order we have to find $x=y$ and $x=\frac{y}2{}$ as a functional limits for $dx$ but I do not know how determine number limits for $dy$

Plot does not make things clear:

How should I handle?

Answer 1

Hint:

complete the region and the plot becomes helpful.

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We see that for $0<y<2$ $x$ we have $\frac{y}{2}<x<y$ and for $2<y<4$ we have $\frac{y}{2}<x<2$

Answer 2

$\int_0^2\mathrm{d}x\int_{x}^{2x}f(x,y)\,\mathrm{d}y=\int_0^2\mathrm{d}y\int_{y/2}^{y}f(x,y)\,\mathrm{d}x+\int_2^4\mathrm{d}y\int_{y/2}^{2}f(x,y)\,\mathrm{d}x.$

Answer 3

Hint: the domain of the integral is y-simple but not x-simple. If you want to reverse the integration order you have to divide the domain into a union of x-simples subdomains and then using the additive property of the integral with respect to disjoint regions.