# Iterated Integrals bounded by curves

## Main Question or Discussion Point

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

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

Hurkyl
Staff Emeritus
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?