\[ \large \int _{ 0 }^{ 2 }{ \lceil x \rceil \, d \lfloor x \rfloor } \]

Evaluate the **Riemann-Stieltjes** integral above, using the definition given in the link.

Enter 666 if you come to the conclusion that this integral fails to exist. \[ \]

**Bonus** What if we use the definition of the Riemann-Stieltjes integral found here (Definition 6.7.2)

**Notations**: \( \lfloor \cdot \rfloor \) denotes the floor function and \( \lceil \cdot \rceil \) denotes the ceiling function.

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