Change the order of Integrtation

1. The problem statement, all variables and given/known data
Change the order of integration and perform the integration.

$$\int_0^2\int_{2x}^{4x-x^2} dydx$$
2. Relevant equations

3. The attempt at a solution
I've tried changing it to this but I end up with the wrong answer..

$$0<=y<=2, \sqrt{4-y}+2 <=x<=\frac{1}{2}y$$
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 Blog Entries: 1 Recognitions: Gold Member Science Advisor Staff Emeritus Sketch the region, what does it look like to you? Have you met polar coordinates yet?
 I've sketched it, it's looks a little cut of a parabola.. I can't see how I can describe it using polar coordinates :/

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Change the order of Integrtation

Nevermind, it seems that polar coordinates are not nesscary, but you should reconsider your limits. Lets look at how the region is bounded. We have;

$$0\leq x \leq 2$$

$$0\leq y \leq 4$$

$$x\leq \frac{1}{2}y \equiv y \geq 2x$$

$$y \leq 4x-x^2$$

Would you agree?
 Yes, I do agree. What I did was solve the last equation and so I got $$0<=y<=2, \sqrt{4-y}+2 <=x<=\frac{1}{2}y$$ .. Still can't see what I'm doing wrong. I'm kinda slow
 Blog Entries: 1 Recognitions: Gold Member Science Advisor Staff Emeritus You've chosen the wrong solution to your equation, $y=4x-x^2$ has two solutions; $$x = 2\pm\sqrt{4-y}$$ Now, your original equation $y=4x-x^2$ goes through the point (x,y)=(0,0), therefore, your new solution must also go through the same point if it is to describe the same region. Which of your solutions goes through the origin?
 Ah, darn I should have seen that! Thank you for being so patient!

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