Let 
be a positive real number. The parabolas defined by 
and 
intersect at the points 
and 
.
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*}](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-9c0b97021c28b4b975a1450142666887_l3.png "Rendered by QuickLaTeX.com")
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*}](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-1eb66d64fec541779631810806101c92_l3.png "Rendered by QuickLaTeX.com")
A site of university mathematics
Let be a positive real number. The parabolas defined by and intersect at the points and .
Prove that the area enclosed by the two curves is constant. Explain why.
Solution
First of all we note that
Hence,