Given a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full".
A full example given in an exam:
A long trough whose cross-section is parabolic is $1\frac{1}{2}$ metres wide at the top and $2$ metres deep. Find the depth of water when it is half-full.
Best Answer
Put the parabola in standard position. It has equation $b^2y=x^2$ for some $b$ which we won't bother to evaluate.
The area of the parabolic segment up to depth $d$ is given by $$\int_{y=0}^d 2x\,dy.$$ Here $x=by^{1/2}$. So the area is $$\frac{2}{3}bd^{3/2},$$ or more simply $kd^{3/2}$ for some constant $k$. If the full depth is $2$, and $m$ is the depth at half the volume, we have $$km^{3/2}=\frac{1}{2}k\, 2^{3/2}.$$ Solve. We get $m^{3/2}=2^{1/2}$ and therefore $m=2^{1/3}$.
Remark: Note that the width at the top is irrelevant to the calculation. Archimedes already knew this, one and a half millenia "before calculus." All the material needed to solve this problem was in his Quadrature of the Parabola. For details, please see the Wikipedia article.