# You can't find an equation for the cross-sectional shape when given only the formula for volume

Why not?

I'm trying to design a calculus I problem. I want them to eventually prove that the volume of a cone (oblique or even with irregular base) is $\frac{\pi r^2 h}{3}$, I also want them to know what a cone is (and what it isn't) -- so I thought this might be a good starting question. But, I don't know if it goes where I want it to.

Update: Making this more clear... I should have said more!

Imagine each "nose cone" has its vertex at the origin and the perpendicular (from the vertex to the base) is the x-axis. If you have a formula A(x) for the cross-sectional area of the "nose cone" at x, perpendicular to the x-axis then the volume is $\int_0^h A(x)dx$.

So, here we have formulas for the volumes of a few solids from NASA. (Only one is a cone.) Can we recover $A(x)$ from these? No, we can't! Not without assuming that the cones are not oblique or otherwise irregular.

(I'll take this as a sign that this would confuse my students!)

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@a little don: I'm sorry, I don't quite understand the question! – Juan S May 3 '11 at 6:54

@a little don: what is $x$? – Fabian May 3 '11 at 7:15
@a little don: $V'=A$, no matter if you write it as $V'(x)=A(x)$ or $V'(h)=A(h)$... – Hans Lundmark May 3 '11 at 7:19