# Interpreting “lying on the parabolas”

Question 49 of chapter 30 of Schaum's calculus is:

The section of a certain solid cut by any plane perpendicular to the x axis is a square with the ends of a diagonal lying on the parabolas $y^2=4x$ and $x^2=4y$ Find its volume.

I don't have a problem integrating the cross sectional area across the x-axis. But I do not understand what this solid's cross sectional square's sides will described as.

What does the author mean by "..with the ends of a diagonal lying on the parabolas.."?

For reference the published solution is $\frac{144}{35}$.

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## 1 Answer

The square cross-sections are oriented with their sides at 45° angles to the $xy$- and $xz$-planes. One vertex is on $y^2=4x$, the opposite vertex is on $x^2=4y$, the other two vertices are above and below the midpoint of the segment (diagonal) joining those two vertices.

edit: a visualization of the solid:

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Ok. The answer is $A(x) = 2x - \frac{x^{5/2}}{2} + \frac{x^4}{32}$. Integrating that $A(x)$ from x=0 to x=4 yields the correct answer. – bobobobo Mar 30 '11 at 22:03
Out of curiosity, how do you get Mathematica to make that? – Ben Alpert Mar 31 '11 at 0:55
@Ben Alpert: Rough outline: the 3d graph is a parametric plot with parameters $x$ (the $x$-coordinate along each curve) and $t$, using $t$ from 0 to 1 to generate the 4 line segments that make up the sides of the square; given the 3d graph, you can set a viewpoint from which to view it, so animate over moving the viewpoint along a circle (which I did by rotating one point about another point in the middle of the figure by a varying angle). If you want, I'll post the code. – Isaac Mar 31 '11 at 0:59