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Mathematics
Calculus
Express the limit as a definite integral
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[QUOTE="andrewkirk, post: 6855006, member: 265790"] You didn't define the lower and upper Darboux sums ##L(f,P), U(f,P)## but let's use the standard definitions from [URL='https://en.wikipedia.org/wiki/Riemann_integral#Riemann_sum']here[/URL]. The relationship follows automatically from the definition since ##t_i\in[x_i,x_{i+1}]## gives us: $$\inf_{t\in [x_i,x_{i+1}]} f(t) \le f(t_i) \le \sup_{t\in [x_i,x_{i+1}]} f(t)$$ We then sum the inequality over ##i## to get the result. BTW the definition of the Riemann integral in the OP is no good, as it requires the axiom of choice (because it requires choosing a set of tags in every one of the uncountably infinitely many possible partitions), and Riemann integrability does not need that axiom. See the second definition in [URL='https://en.wikipedia.org/wiki/Riemann_integral#Riemann_integral']this section [/URL]for an intuitive, workable and robust definition of a Riemann integral. [/QUOTE]
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Express the limit as a definite integral
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