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A curve with Cartesian equation $y = f(x)$ passes through the origin. Lines drawn parallel to the coordinate axes through an arbitrary point of the curve form a rectangele with two sides on the axes. The curve divides every such rectangle into two regions $A$ and $B$, one of which has an area equal to $n$ times the other. What are all possible functions $f$ with this property?

What possible functions $f$ exist such that the two regions $A$ and $B$ have the property that, when rotated about the $x$-axis, they sweep out solids one of which has a volume $n$ times that of the other?

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For the area, we may express the condition as

$$\int_0^x dx' \, f(x') = n \left [ x f(x) - \int_0^x dx' \, f(x')\right]$$

which implies that

$$\int_0^x dx' \, f(x') = \frac{n}{n+1} x f(x)$$

Differentiating, we get

$$n x f'(x) = f(x)$$

which has solution

$$f(x) = A x^{1/n}$$

for some constant $A > 0$.

For the volume problem, the analysis is very similar:

$$\int_0^x dx' \, f(x')^2 = n \left [ x f(x)^2 - \int_0^x dx' \, f(x')^2\right]$$

The solution here is

$$f(x) = A x^{1/(2 n)}$$

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  • $\begingroup$ Fantastic! Thank you very much! $\endgroup$ – Saaqib Mahmood Aug 2 '13 at 12:01
  • $\begingroup$ @SaaqibMahmuud: You're welcome. Please remember to accept a solution by checking on the checkmark to the left if you find the solution useful. $\endgroup$ – Ron Gordon Aug 2 '13 at 12:03

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