Double integral area problem

**gralla55** · October 5th 2011, 11:44 AM

"Use double integrals to calculate the area of the region bounded by y=2x, x=0 and y=1-2x-x^2!

The only thing I've done so far is to create a scetch of the region, and solve for the x-value the to functions intersect. I'm not completely sure how to set up the integrals or in what order. Since the region is vertically simple, it would make sense to integrate with respect to x first, but then I would have to solve y=1-2x-x^2 in terms of x, which can't be done (or can it?)... So the best thing I've come up with is to split the region into 2, and determine the area of each by integrating with respect to y first.

Any help? Thanks!

**MonroeYoder** · October 6th 2011, 08:02 AM

$y=1-2x-x^2=1-2x-x^2+2-2$ (add zero in the form +2-2)

$y-2=-1-2x-x^2$

$2-y=x^2+2x+1=(x+1)^2$

$x+1=\pm\sqrt{2-y}$

$x=\pm\sqrt{2-y}-1$

In your case you only care about $x>0$ so forget about $x=-\sqrt{2-y}-1$

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