# Volumes using Cross Sections (Integrals)

I have a question that states as follows:

"A solid lies between planes perpendicular to the $x$-axis at $x=0$ and $x=9$. The cross sections perpendicular to the axis on the interval $0 \leq x \leq 9$ are squares with diagonals that run from the parabola $y=-2\sqrt{x}$ to $y=2\sqrt{x}$. Find the volume of the solid."

My difficulty is this: I can plug numbers into an integration formula that I know and get the correct answer, but I can't seem to picture or draw this problem geometrically, and this bothers me greatly because it suggests that I do not really understand the concept as a whole. I had questions like these on my calculus I final, missed them, and didn't get the grade that I was going for as a result. I am going to take calculus II and III, so I want to make sure that I have this stuff down and that it does not hinder me.

I am able to conceptualize and graph the parabolic part, but it is the statement about the diagonal that I am at a loss to picture.

Can anybody give me any pointers as to what I may be missing?

Thank you so much.

Let $S(x)$ means area of cross section for $x$, where $x \in [0,9]$, we know that cross sections are squares with diagonal from $(x,2\sqrt{x})$ to $(x,-2\sqrt{x})$, so the area of thesquare for $x$ is $S(x)=\frac{(2\sqrt{x}+2\sqrt{x})^2}{2}=8x$. Now by Fubini theorem:
$V=\int_{0}^{9} S(x) dx=\int_{0}^{9} 8x dx=4*9^2-0^2=324$