# That's a lot of things to average

Calculus Level 5

On $$x \in [a, b]$$, the area under $$y = x^2$$ is always 5. That is, for any $$(a, b)$$ you choose, $$a$$ and $$b$$ satisfy $$\displaystyle \int_{a}^{b} x^2 \, dx = 5$$ .

On $$x \in [0, 100 ]$$, there is an infinite number of possibilities for $$[a, b]$$ that satisfy the above conditions. Then, you can get an infinite number of distances between $$a$$ and $$b$$ depending on the values of $$a$$ and $$b$$.

If the average distance between $$a$$ and $$b$$ given all the above conditions is $$\mathfrak{D}$$, find $$\left \lceil 1000 \mathfrak{D} \right \rceil$$.

×