Change the order of Integrtation

[Dafe]

Homework Statement

Change the order of integration and perform the integration.

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

Homework Equations

The Attempt at a Solution

I've tried changing it to this but I end up with the wrong answer..

$$0<=y<=2, \sqrt{4-y}+2 <=x<=\frac{1}{2}y$$

 

[Hootenanny]

Sketch the region, what does it look like to you? Have you met polar coordinates yet?

 

[Dafe]

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 :/

 

[Hootenanny]

Nevermind, it seems that polar coordinates are not nesscary, but you should reconsider your limits. Let's 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?

 

[Dafe]

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

 

[Hootenanny]

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?

 

[Dafe]

Ah, darn I should have seen that! Thank you for being so patient!

 

[Hootenanny]

Ah, darn I should have seen that! Thank you for being so patient!

Nah, its no problem, I didn't spot it at first until I started sketching the region