# Thread: Double integral over strange region

1. ## Double integral over strange region

The parabolas $\displaystyle x = -y^2, x = 2y - y^2, x = 2 - y^2 - 2y$ divide the plane into 7 regions, of which only one is bounded. Let this region be A. Find a change of variables such that the first two parabolas become u=0 and v=0. Evaluate the double integral $\displaystyle \int_{A} x dx dy$.

2. I should clarify - I can find the Jacobian factor and the "new" integrand in terms of u and v. The only problem is that I don't know what the region should be (ie the limits of the u and v integrals). In fact I'm not too sure how to desribe the region on x and y coordinates (ie I'm not sure what the limit of the integrals would have been before the change of variables anyway).

3. A way to see it better would to graph the 3 curves. Check out the region 0<y<1 , -2<x<2.

So the integration in that region would be, in x and y coordinates:

$\displaystyle \int_{0 }^{ \frac{1 }{2 } } \int_{ -y^{2}}^{2y-y^{2} }xdxdy + \int_{ \frac{ 1}{ 2} }^{ 1 } \int_{ -y^{2}}^{2-y^{2}-2y }xdxdy$

Darnit! For some reason it wont let me edit my own posts. My integrals are slightly wrong, they should be $\displaystyle dxdy$ instead of $\displaystyle dydx$

[Fixed. -K.]