# Double integral over a region

1. Apr 12, 2015

### nuuskur

1. The problem statement, all variables and given/known data
$\iint\limits_D x{\rm{d}}x{\rm{d}}y$ where $x = \sqrt{2y - y^2}, y = \sqrt{2x - x^2}$

2. Relevant equations

3. The attempt at a solution
I have figured out the region in question:
But how do I get the limits of integration?

Ah, perhaps..
$$\int_0^1 \int_{1-\sqrt{1-y^2}}^{\sqrt{2y-y^2}} x {\rm{d}}x{\rm{d}}y$$
In the inner integral I consider what X is bound by and then the 2nd integration would be just from 0 to 1 since that's what y is bound by. Not really worried about the actual calculation itself, but the most challenging bit is figuring out the bounds.

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Last edited: Apr 12, 2015
2. Apr 12, 2015

### BruceW

yep. those bounds look good. I get the same ones at least :)

edit: you could also do the integration in the opposite order (i.e. dy first), but I think that way is more time-consuming. and of course, in that method, the limits would look similar, but would have x instead of y.

Last edited: Apr 12, 2015
3. Apr 12, 2015

### LCKurtz

@nuuskur: Of course, if you were "worried" about evaluating the integral, it would be wise to convert the equations to polar coordinates and set it up and work it that way.