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$f(x)= 1/2$ if x is rational $sin(x)$ if x is irrational.

Im not sure if im doing it correctly: $L(P,f)=(3-\frac{-\pi}{6})(sin(\frac{-\pi}{6}))+(2\pi-3)(sin(2\pi))$

Another question. If I have a function f(x) = constant, how would I choose the lower and upper bounds.

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What is $P\{\cdot;\cdot;\cdot\}$ supposed to be? How is $L$ defined?

In any interval $[a;b]$ or $(a;b]$ or $[a;b)$ or $(a;b)$ the lower is $Min\{\frac{1}{2}\}\cup\{\sin(x):x\in[a;b]\}$, which will w.l.o.g upon appropriate adjustments of the intervals be equal to $Min\{\frac{1}{2};\sin(a);\sin(b)\}$.

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