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Constant area

Let a be a positive real number. The parabolas defined by y_1=ax^2 and y_2^2=ax intersect at the points \mathrm{O} and \mathrm{A}.

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Prove that the area enclosed by the two curves is constant. Explain why.

Solution

First of all we note that

    \begin{align*} y_1 = y_2 &\Leftrightarrow \left ( ax^2 \right )^2 = ax \\ &\Leftrightarrow a^2 x^4 = ax \\ &\!\!\!\!\!\overset{a>0}{\Leftarrow \! =\! =\! \Rightarrow } a x^4 - x =0 \\ &\Leftrightarrow x \left ( ax^3 -1 \right ) =0 \\ &\Leftrightarrow \left\{\begin{matrix} x & = & 0\\ x &= & \sqrt[3]{\frac{1}{a}} \end{matrix}\right. \end{align*}

Hence,

    \begin{align*} \mathrm{E}\left ( \Omega \right ) &= \int_{0}^{\sqrt[3]{1/a}} \left | ax^2 - \sqrt{ax} \right |\, \mathrm{d}x\\ &=\int_{0}^{\sqrt[3]{1/a}} \left ( \sqrt{ax} - ax^2 \right )\, \mathrm{d}x \\ &=\frac{2}{3} - \frac{1}{3} \\ &= \frac{1}{3} \end{align*}

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