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Explicit formula of f

In the following figure the function f is continuous and 1-1. For every point \mathrm{P} on the curve y=2x^2 the areas of \mathrm{A} and \mathrm{B} are equal.

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You are asked to find an explicit formula for f.

Solution

We are expanding the above figure so that it looks like this;

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The area \mathrm{A} is given by

    \[\mathrm{A} = \int_{0}^{\mathbf{b}}2x^2 \, \mathrm{d} x - \int_{0}^{\mathbf{b}} x^2 \, \mathrm{d}x = \int_{0}^{\mathbf{b}} x^2 \, \mathrm{d}x = \frac{\mathbf{b}^3}{3}}\]

On the other hand we have that

    \begin{align*} \mathbf{b} f(a) &= \int_{0}^{\mathbf{b}} x^2 \, \mathrm{d}x + \mathrm{A} + \mathrm{B} + \left ( a f(a) - \int_{0}^{\mathbf{a}} f(x) \, \mathrm{d}x \right ) \\ &\!\!\!\!\!\!\!\!\!\!\!\!\overset{\mathrm{A} = \mathrm{B} \; , \; f(a)=2b^2}{=\! =\! =\! =\! =\! =\!=\! =\!=\! =\!} \int_{0}^{\mathbf{a}} f(x) \, \mathrm{d}x = 2\mathbf{a} \mathbf{b}^2 - \mathbf{b^3} \end{align*}

Differentiating with b as a variable and a(b) dependent to b we get that \mathbf{b} = \frac{3a}{4}. Hence,

    \[f(x) = \frac{32x^2}{9}\]

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