# How to set up this double integral?

I have the double integral ∫∫x dA bounded by the curves y=1, y=-x, and y = √x

I drew out the graph, but I'm having trouble determining what the bounds for the integrals are. Is x from -y to y^2, and y from 0 to 1, or is x from 0 to 1 and y from -x to √x (or vice versa)?

• The square root of x is not x^2! Try again, and I'm sure you'll get it... Dec 14 '11 at 16:02
• Whoops, I entered it wrong on wolfram alpha to generate that graph image. On my own paper, I have the right graph, but the question still remains. Removed the graph image to avoid confusion. Dec 14 '11 at 16:04
• Once that is sorted out, split the integral into two integrals, at $x =0$. The upper function is "1" throughout, while the lower function is $-x$ on [-1,0] and $\sqrt x$ on (0,1] Dec 14 '11 at 16:05
• You could also integrate $dy$, using only one integral... Dec 14 '11 at 16:06
• Also, once you figure it out, you are encouraged to answer your own question. Obviously this isn't possible in many situations, but I suspect you can take the hints and roll with them. Dec 14 '11 at 16:15

$\int_0^1 \int_{-y}^{y^2} dx \ dy$ is what you would get with horizontal "representative rectangles". So in this case, the inner integration gives:

$\int_0^1 (y^2 - (-y)) dy$, which is the integral I hinted at.

If you want to integrate dx,

$A = \int_{-1}^0 (1 - (-x)) dx + \int_0^1 (1 - \sqrt x) dx$

• Regarding you doubts that y = 1 is the "upper" function in the dx approach, I'd suggest that you redraw the region and see that no other region is bounded by all three equations. Dec 14 '11 at 16:27
• Thanks, I really only needed the bounds to practice setting up integrals with curves given, but you confirmed that my initial guess was correct Dec 14 '11 at 16:35
• I was the victim of serial downvoting on 5 January 2012. Ask me anything. Feb 14 '12 at 0:17
• @TheChaz What happened!?!??! Feb 21 '12 at 2:01
• @TheChaz =) That was my reaction to that. I discovered that many people tend to get a little worked up when discussing maths, kind of looking for the other's mistake but not providing their insight, correction or proof. I guess it's an aspect to work on. Feb 21 '12 at 3:41