Iterated Integrals bounded by curves

[braindead101]

Evaluate \int\int_{Q}\left(1 - x^{3}\right)y^{2} dA where Q is the region bounded by y=x^2 and x = y^2

So I have drew the graphs of y=x^2 and x=y^2 and found that they intersect at (0,0) and (1,1). Now I am confused what to replace Q with, but I think it should be this: please tell me if I am incorrect in my selection.

\int^{1}_{0}\int_{\sqrt{y}}^{y^{2}}\left(1 - x^{3}\right)y^{2} dx dy

or should I be integrating w.r.t y first? also have I mixed up the y^2 and the sqrt(y) in the limit of integration?

 

[EnumaElish]

dxdy is correct, but sqrt(y) > y^2.

 

[braindead101]

how would i know this for future reference, i am having serious trouble with the limit of integration part.

 

[Hurkyl]

Well, it's clear that either \sqrt{y} < y^2 for every y in that interval, or y^2 < \sqrt{y} for every y in that interval, correct?

So, if you try one actual value of y in that interval...


A little algebra would solved it too: what happens if you manipulate that inequality to put all of the y's on the same side?