Textbook Problem: Replace the given iterated integral by an equivalent one with the order of integration reversed.

$\displaystyle \int_0^2{\int_0^{x^2}{x^3 y \, dy} \, dx}$

I came up with the answer:

$\displaystyle \int_0^4{\int_{\sqrt{y}}^2{x^3 y \, dx} \, dy}$

The answer section of the textbook says:

$\displaystyle \int_0^4{\int_{\sqrt{y}}^y{x^3 y \, dx} \, dy}$

The upper limit of the interval of integration is different. Did I make a mistake or did the textbook make a mistake?

I would like to make sure that I am doing this stuff correctly.