Double Integral Doubts: Understanding Regions and Order

[Telemachus]

Homework Statement

Hi there, I've got this doubt about a double integral. I have this region: $$\displaystyle\int_{-1}^{2}\displaystyle\int_{-\sqrt[ ]{4-x^2}}^{1-x^2}f(x,y)dydx$$

And the thing is, how this region would look like? Would it look like this?:

The thing is that after the cut between the two curves the order changes, so I think that region would have an opposite sign than the region before.

What you say?

Bye, thanks for posting!

 

[HallsofIvy]

Yes, that's correct. The integral will be equal to the area of the region on the left minus the area of the region on the right.

 

[Telemachus]

Thanks HallsofIvy. It turns confusing since that area will generate a volume under the graph f(x,y). I don't know if it really have any sense that region planted that way.

Mmm now I think that the volume for the last part would be negative (or positive depending on f) and then it would have some sense.

By the way, is this equality right? $$\displaystyle\int_{-3}^{1}\displaystyle\int_{arctg(x)}^{e^x}f(x,y)dydx=\displaystyle\int_{arctg(x)}^{e^x}\displaystyle\int_{-3}^{1}f(x,y)dxdy$$