Volume of Static Solid Using Cross-Sectional Area (Integration)

[Ascendant0]

Homework Statement

The flat base of a solid sits in the xy-plane in the region bounded by the x-axis, the line $y = 8$, and $y = x^3$. Evaluate an integral which represents the volume of this solid if cross-sections taken perpendicular to the x-axis at x are squares

Relevant Equations

$y = 8$
$y = x^3$

Ok, so doing this one direction, with the range of x (0 to 2), I get the top minus the bottom equation of:

$y = 8 - x^3$

Then, since it's squares, this would make it $y^2$. So, integrating gives:

$\int_{0}^{2} (8-x^3)^2 = 82.3$

That seems to be correct. However, I want to make sure I FULLY understand how to do these types of things, so I wanted to evaluate using the y-axis instead. I'm doing something seriously wrong, but can't figure out what.

So, when trying to evaluate across y, I use the equations:

$y = 8$ and $x = y^{1/3}$

Again, since it's squares, taking the integral makes $y^{1/3}$ into $y^{2/3}$ and the resulting integral of:

$\int_{0}^{8} y^{2/3} = 3/5 \ y^{5/3} = 19.2$

And so of course, I know that answer is entirely wrong and I'm setting this up wrong, but I can't figure out where I'm going wrong here. Some help please?

 

[Hill]

since it's squares

The cross-sections taken perpendicular to the y-axis are not squares.

 

[Ascendant0]

The cross-sections taken perpendicular to the y-axis are not squares.

Omg, that should've been glaringly obvious to me, lol. Thank you so much for pointing that out. Now it makes complete sense as soon as you said that and I revisited a 3d image of it. Wow, no wonder it was so far off. I appreciate the help!