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This is a famous calculus problem and is stated like this

Given a barrel with height $h$, and a small radius of $a$ and large radius of $b$. Calculate the volume of the barrel given that the sides are parabolic.

Now I seem to have solved the problem incorrectly because here it seems 2that the volume should be

$ \displaystyle \hspace{1cm} V(a,b,h) = \frac{h\pi}{3}\left(2b^2 + a^2\right)\,. $

Below is my attempt. As in the picture I view the barrel from the side, and try to find a formula for the parabola. So i solve

$ \displaystyle \hspace{1cm} f(x) := A x^2 + B x + C $

given $f(0) = f(h) = a/2$ and $f(h/2) = b/2$. This yields

$ \displaystyle \hspace{1cm} f(x) = \frac{2(a-b)}{h^2} \cdot x^2 - \frac{2(a-b)}{h} \cdot x + \frac{a}{2} $

Using the shell method integrating now gives the volume as

$ \displaystyle \hspace{1cm} V(a,b,h) := \pi \int_0^h \bigl[f(x)\bigr]^2\,\mathrm{d}x = \frac{\pi}{60} \cdot h (a+2b)^2 + \frac{\pi}{30} \cdot h(a^2+b^2) $

Alas according to the formula above this seems incorrect! Where is my mistake?

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3 Answers

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Let $k=h/2$, and put the origin in the middle, where symmetry asks it to be.

Then the equation of the upper parabola is $$y=b-\frac{b-a}{k^2}x^2.$$ The integral of $\pi y^2\,dx$ from $0$ to $k$ is $$\pi k\left(b^2-\frac{2}{3}(b-a)b+\frac{1}{5}(b-a)^2\right).$$ This simplifies to $$\frac{\pi k}{15}(3a^2+4ab+8b^2)$$ Replace $k$ by $h/2$ and multiply by $2$.

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Note that in their case $a$, $b$ are radiuses, not diameters. Another important thing to say is that the formula from that external site is not exact!

For simplicity, take $h=1$. Let's define our function on $[-1/2,1/2]$ as $f(x) = 4(a-b)x^2+b$. We write

$$\frac{V}{\pi}= 2\int _{0}^{1/2}(4(a-b)x^2+b)^2 \mathrm dx= 16 (a-b)^2 \cdot \frac{1}{5\cdot 16}+ 8 b(a-b)\cdot \frac{1}{3\cdot 4} + b^2$$ $$= (a-b)^2/5 + 2 b(a-b)/3 + b^2.$$

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Volume of a barrel

V = 0.5 * πQuadrat * (π/18 + 1/6) * (r1Quadrat + r2Quadrat) * h parabolic bend
V = 0.5 * πQuadrat * (π/17 + 1/6) * (r1Quadrat + r2Quadrat) * h elliptic bend

Best regards Hans-Jürgen Gläsel

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