Stranded on an Island

Calculus Level 3

You are stranded on an island when a boat full of mathematicians floats past. You want to be rescued so you put this integral on a large sign:

010x+22dx \int_0^{10} {\lfloor x+2 \rfloor}^2 \, dx

What is the value of this integral?

Bonus: And why were you rescued?

Notation: \lfloor \cdot \rfloor denotes the floor function.

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