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Consider the functions $x^2$ and $ax$ for $x\ge0$. Specifically, consider the average area between the two curves as $a$ modulates between $0$ and $k$ and refer to it as $\hat a$. Which non-zero value of $k$ satisfies the condition that $\hat a=x^2$?

Although I intend to answer my own question, I would appreciate reading other solutions and viewpoints for this problem. I am trying to become better at mathematical writing so viewing other solutions would be greatly beneficial to myself and others. Additionally, if you choose not to read my answer before coming up with your own, this could potentially become quite interesting as the methodology could vary widely.

In case the wording of the problem is confusing, I modeled a picture of the area when $a=1$.

Area between $x^2$ and $x$

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Find the intersections of $x^2$ and $ax$ to be $x^2-ax=0\iff x(x-a)=0$, and thus our bound is $[0,a]$. We can then calculate the area for any specific $a$ as $\int_0^a\left(x^2-ax\right)dx=\frac{a^3}3-a\frac{a^2}{2}=\frac {a^3} 6$. Finally we calculate $\hat a$ as $\frac 1 k\int_0^k\frac {a^3} 6\ da=\frac {k^3}{24}$ and thus the solution $k=24$ since $24^2=\frac {24^3}{24}$.

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