Does the volume with cross sections perpendicular to $x$-axis equal the cross sections perpendicular to $y$-axis?

Question

I've been working on this problem: Find the volume of the solid whose base is the region bounded between the curve $y=x^2$ and the $x$-axis from $x=0$ to $x=2$ and whose cross sections taken perpendicular to the $x$-axis are squares. $$\int_0^2 (x^2)^2 dx = \frac{32}5,$$ but when I tried to take the cross section perpendicular to the $y$-axis with squares I get $$\int_0^4 (2-\sqrt{y})^2 dy = \frac{8}3.$$ Shouldn't both those areas be equal? Aren't I covering the same amount of area in both integrals? What am I doing wrong? Or they shouldn't be equal? Why not? Thanks!