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 \( \left \lceil 1000 \mathfrak{D} \right \rceil \).

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